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diff --git a/build/pkgs/giac/patches/pari_2_15.patch b/build/pkgs/giac/patches/pari_2_15.patch
new file mode 100644
index 0000000000..d2900a5ffc
--- /dev/null
+++ b/build/pkgs/giac/patches/pari_2_15.patch
@@ -0,0 +1,21 @@
+ANYARG patch
+
+diff --git a/src/pari.cc b/src/pari.cc
+index 76ce8e1..50d08ab 100644
+--- a/src/pari.cc
++++ b/src/pari.cc
+@@ -40,6 +40,13 @@ using namespace std;
+
+ #ifdef HAVE_LIBPARI
+
++// Anyarg disappeared from PARI 2.15.0
++#ifdef __cplusplus
++# define ANYARG ...
++#else
++# define ANYARG
++#endif
++
+ #ifdef HAVE_PTHREAD_H
+ #include <pthread.h>
+ #endif
+
diff --git a/build/pkgs/pari/checksums.ini b/build/pkgs/pari/checksums.ini
index b736feed31..bafd0f36f4 100644
--- a/build/pkgs/pari/checksums.ini
+++ b/build/pkgs/pari/checksums.ini
@@ -1,5 +1,5 @@
tarball=pari-VERSION.tar.gz
-sha1=e01647aab7e96a8cb4922cf26a4f224337c6647f
-md5=922f740fcdf8630b30d63dc76b58f756
-cksum=297133525
+sha1=cba9b279f67d5efe2fbbccf3be6e9725f816cf07
+md5=76d430f1bea1b07fa2ad9712deeaa736
+cksum=1990743897
upstream_url=https://pari.math.u-bordeaux.fr/pub/pari/unix/pari-VERSION.tar.gz
diff --git a/build/pkgs/pari/package-version.txt b/build/pkgs/pari/package-version.txt
index a1a4224dd5..68e69e405e 100644
--- a/build/pkgs/pari/package-version.txt
+++ b/build/pkgs/pari/package-version.txt
@@ -1 +1 @@
-2.13.3
+2.15.0
diff --git a/src/doc/de/tutorial/tour_numtheory.rst b/src/doc/de/tutorial/tour_numtheory.rst
index a012234c99..e3149fe949 100644
--- a/src/doc/de/tutorial/tour_numtheory.rst
+++ b/src/doc/de/tutorial/tour_numtheory.rst
@@ -157,7 +157,7 @@ implementiert.
Univariate Quotient Polynomial Ring in a over Rational Field with modulus
x^3 + x^2 - 2*x + 8
sage: K.units()
- (3*a^2 + 13*a + 13,)
+ (-3*a^2 - 13*a - 13,)
sage: K.discriminant()
-503
sage: K.class_group()
diff --git a/src/doc/en/tutorial/tour_numtheory.rst b/src/doc/en/tutorial/tour_numtheory.rst
index 3064d100e2..075e0ac0ad 100644
--- a/src/doc/en/tutorial/tour_numtheory.rst
+++ b/src/doc/en/tutorial/tour_numtheory.rst
@@ -157,7 +157,7 @@ NumberField class.
Univariate Quotient Polynomial Ring in a over Rational Field with modulus
x^3 + x^2 - 2*x + 8
sage: K.units()
- (3*a^2 + 13*a + 13,)
+ (-3*a^2 - 13*a - 13,)
sage: K.discriminant()
-503
sage: K.class_group()
diff --git a/src/doc/es/tutorial/tour_numtheory.rst b/src/doc/es/tutorial/tour_numtheory.rst
index a1f7d1a87b..48e5376cfe 100644
--- a/src/doc/es/tutorial/tour_numtheory.rst
+++ b/src/doc/es/tutorial/tour_numtheory.rst
@@ -140,7 +140,7 @@ Varios métodos relacionados están implementados en la clase ``NumberField``::
Univariate Quotient Polynomial Ring in a over Rational Field with modulus
x^3 + x^2 - 2*x + 8
sage: K.units()
- (3*a^2 + 13*a + 13,)
+ (-3*a^2 - 13*a - 13,)
sage: K.discriminant()
-503
sage: K.class_group()
diff --git a/src/doc/fr/tutorial/tour_numtheory.rst b/src/doc/fr/tutorial/tour_numtheory.rst
index 871092f5fa..d1b2fee883 100644
--- a/src/doc/fr/tutorial/tour_numtheory.rst
+++ b/src/doc/fr/tutorial/tour_numtheory.rst
@@ -159,7 +159,7 @@ dans la classe NumberField.
Univariate Quotient Polynomial Ring in a over Rational Field with modulus
x^3 + x^2 - 2*x + 8
sage: K.units()
- (3*a^2 + 13*a + 13,)
+ (-3*a^2 - 13*a - 13,)
sage: K.discriminant()
-503
sage: K.class_group()
diff --git a/src/doc/ja/tutorial/tour_numtheory.rst b/src/doc/ja/tutorial/tour_numtheory.rst
index 47af68c862..4d4ed52d50 100644
--- a/src/doc/ja/tutorial/tour_numtheory.rst
+++ b/src/doc/ja/tutorial/tour_numtheory.rst
@@ -161,7 +161,7 @@ Sageには :math:`p` \-進数体も組込まれている．
Univariate Quotient Polynomial Ring in a over Rational Field with modulus
x^3 + x^2 - 2*x + 8
sage: K.units()
- (3*a^2 + 13*a + 13,)
+ (-3*a^2 - 13*a - 13,)
sage: K.discriminant()
-503
sage: K.class_group()
diff --git a/src/doc/pt/tutorial/tour_numtheory.rst b/src/doc/pt/tutorial/tour_numtheory.rst
index 6371b491ea..a3dc973a93 100644
--- a/src/doc/pt/tutorial/tour_numtheory.rst
+++ b/src/doc/pt/tutorial/tour_numtheory.rst
@@ -157,7 +157,7 @@ NumberField.
Univariate Quotient Polynomial Ring in a over Rational Field with modulus
x^3 + x^2 - 2*x + 8
sage: K.units()
- (3*a^2 + 13*a + 13,)
+ (-3*a^2 - 13*a - 13,)
sage: K.discriminant()
-503
sage: K.class_group()
diff --git a/src/doc/ru/tutorial/tour_numtheory.rst b/src/doc/ru/tutorial/tour_numtheory.rst
index 652abfbc99..a985d49fbd 100644
--- a/src/doc/ru/tutorial/tour_numtheory.rst
+++ b/src/doc/ru/tutorial/tour_numtheory.rst
@@ -150,7 +150,7 @@ Sage содержит стандартные функции теории чис
Univariate Quotient Polynomial Ring in a over Rational Field with modulus
x^3 + x^2 - 2*x + 8
sage: K.units()
- (3*a^2 + 13*a + 13,)
+ (-3*a^2 - 13*a - 13,)
sage: K.discriminant()
-503
sage: K.class_group()
diff --git a/src/sage/arith/misc.py b/src/sage/arith/misc.py
index e57076646f..fec75d07c1 100644
--- a/src/sage/arith/misc.py
+++ b/src/sage/arith/misc.py
@@ -1465,13 +1465,13 @@ def divisors(n):
sage: K.<a> = QuadraticField(7)
sage: divisors(K.ideal(7))
- [Fractional ideal (1), Fractional ideal (-a), Fractional ideal (7)]
+ [Fractional ideal (1), Fractional ideal (a), Fractional ideal (7)]
sage: divisors(K.ideal(3))
[Fractional ideal (1), Fractional ideal (3),
- Fractional ideal (-a + 2), Fractional ideal (-a - 2)]
+ Fractional ideal (a - 2), Fractional ideal (a + 2)]
sage: divisors(K.ideal(35))
- [Fractional ideal (1), Fractional ideal (5), Fractional ideal (-a),
- Fractional ideal (7), Fractional ideal (-5*a), Fractional ideal (35)]
+ [Fractional ideal (1), Fractional ideal (5), Fractional ideal (a),
+ Fractional ideal (7), Fractional ideal (5*a), Fractional ideal (35)]
TESTS::
@@ -2569,7 +2569,7 @@ def factor(n, proof=None, int_=False, algorithm='pari', verbose=0, **kwds):
sage: K.<i> = QuadraticField(-1)
sage: factor(122 - 454*i)
- (-3*i - 2) * (-i - 2)^3 * (i + 1)^3 * (i + 4)
+ (-i) * (-i - 2)^3 * (i + 1)^3 * (-2*i + 3) * (i + 4)
To access the data in a factorization::
diff --git a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
index cdfc11a9e5..b6e1280d6e 100644
--- a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
+++ b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
@@ -7825,9 +7825,9 @@ class DynamicalSystem_projective_field(DynamicalSystem_projective,
sage: f = DynamicalSystem_projective([x^2 + QQbar(sqrt(3))*y^2, y^2, QQbar(sqrt(2))*z^2])
sage: f.reduce_base_field()
Dynamical System of Projective Space of dimension 2 over Number Field in a with
- defining polynomial y^4 - 4*y^2 + 1 with a = 1.931851652578137?
+ defining polynomial y^4 - 4*y^2 + 1 with a = -0.5176380902050415?
Defn: Defined on coordinates by sending (x : y : z) to
- (x^2 + (a^2 - 2)*y^2 : y^2 : (a^3 - 3*a)*z^2)
+ (x^2 + (-a^2 + 2)*y^2 : y^2 : (a^3 - 3*a)*z^2)
::
diff --git a/src/sage/ext_data/pari/simon/ellQ.gp b/src/sage/ext_data/pari/simon/ellQ.gp
index 420af8f6a2..65e8386779 100644
--- a/src/sage/ext_data/pari/simon/ellQ.gp
+++ b/src/sage/ext_data/pari/simon/ellQ.gp
@@ -40,7 +40,7 @@
gp > \r ellcommon.gp
gp > \r ellQ.gp
- The main function is ellrank(), which takes as an argument
+ The main function is ellQ_ellrank(), which takes as an argument
any elliptic curve in the form [a1,a2,a3,a4,a6]
the result is a vector [r,s,v], where
r is a lower bound for the rank,
@@ -50,7 +50,7 @@
Example:
gp > ell = [1,2,3,4,5];
- gp > ellrank(ell)
+ gp > ellQ_ellrank(ell)
%1 = [1, 1, [[1,2]]
In this example, the rank is exactly 1, and [1,2] has infinite order.
@@ -92,7 +92,7 @@
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
Explications succintes :
- La fonction ellrank() accepte toutes les courbes sous la forme
+ La fonction ellQ_ellrank() accepte toutes les courbes sous la forme
[a1,a2,a3,a4,a6]
Les coefficients peuvent etre entiers ou non.
L'algorithme utilise est celui de la 2-descente.
@@ -100,7 +100,7 @@
Il suffit de taper :
gp > ell = [a1,a2,a3,a4,a6];
- gp > ellrank(ell)
+ gp > ellQ_ellrank(ell)
Retourne un vecteur [r,s,v] ou
r est le rang probable (c'est toujours une minoration du rang),
@@ -110,7 +110,7 @@
Exemple :
gp > ell = [1,2,3,4,5];
- gp > ellrank(ell)
+ gp > ellQ_ellrank(ell)
%1 = [1, 1, [[1,2]]
Ici, le rang est exactement 1, et le point [1,2] est d'ordre infini.
@@ -1571,12 +1571,12 @@ if( DEBUGLEVEL_ell >= 4, print(" end of ell2descent_gen"));
print("rank(E/Q) >= ",m1)
);
}
-{ellrank(ell,help=[]) =
+{ellQ_ellrank(ell,help=[]) =
\\ Algorithm of 2-descent on the elliptic curve ell.
\\ help is a list of known points on ell.
my(urst,urst1,den,eqell,tors2,bnf,rang,time1);
-if( DEBUGLEVEL_ell >= 3, print(" starting ellrank"));
+if( DEBUGLEVEL_ell >= 3, print(" starting ellQ_ellrank"));
if( #ell < 13, ell = ellinit(ell));
\\ kill the coefficients a1 and a3
@@ -1630,7 +1630,7 @@ if( DEBUGLEVEL_ell >= 1, print(" Elliptic curve: Y^2 = ",eqell));
));
rang[3] = ellchangepoint(rang[3],ellinverturst(urst));
-if( DEBUGLEVEL_ell >= 3, print(" end of ellrank"));
+if( DEBUGLEVEL_ell >= 3, print(" end of ellQ_ellrank"));
return(rang);
}
@@ -2106,13 +2106,13 @@ if( DEBUGLEVEL_ell >= 3, print(" end of ell2descent_viaisog"));
{
\\ functions for elliptic curves
addhelp(ell2descent_complete,
- "ell2descent_complete(e1,e2,e3): Performs a complete 2-descent on the elliptic curve y^2 = (x-e1)*(x-e2)*(x-e3). See ?ellrank for the format of the output.");
+ "ell2descent_complete(e1,e2,e3): Performs a complete 2-descent on the elliptic curve y^2 = (x-e1)*(x-e2)*(x-e3). See ?ellQ_ellrank for the format of the output.");
addhelp(ell2descent_gen,
- "ell2descent_gen((E,bnf,k=1,help=[]): E is a vector of the form [0,A,0,B,C], (or the result of ellinit of such a vector) A,B,C integers such that x^3+A*x^2+B*x+C; bnf is the corresponding bnfinit(,1); Performs 2-descent on the elliptic curve Ek: k*y^2=x^3+A*x^2+B*x+C. See ?ellrank for the format of the output.");
+ "ell2descent_gen((E,bnf,k=1,help=[]): E is a vector of the form [0,A,0,B,C], (or the result of ellinit of such a vector) A,B,C integers such that x^3+A*x^2+B*x+C; bnf is the corresponding bnfinit(,1); Performs 2-descent on the elliptic curve Ek: k*y^2=x^3+A*x^2+B*x+C. See ?ellQ_ellrank for the format of the output.");
addhelp(ell2descent_viaisog,
- "ell2descent_viaisog(E,help=[]): E is an elliptic curve of the form [0,a,0,b,0], with a, b integers. Performs a 2-descent via isogeny on E. See ?ellrank for the format of the output.");
- addhelp(ellrank,
- "ellrank(E,help=[]): E is any elliptic curve defined over Q. Returns a vector [r,s,v], where r is a lower bound for the rank of E, s is the rank of its 2-Selmer group and v is a list of independant points in E(Q)/2E(Q). If help is a vector of nontrivial points on E, the result might be faster. This function might be used in conjunction with elltors2(E). See also ?default_ellQ");
+ "ell2descent_viaisog(E,help=[]): E is an elliptic curve of the form [0,a,0,b,0], with a, b integers. Performs a 2-descent via isogeny on E. See ?ellQ_ellrank for the format of the output.");
+ addhelp(ellQ_ellrank,
+ "ellQ_ellrank(E,help=[]): E is any elliptic curve defined over Q. Returns a vector [r,s,v], where r is a lower bound for the rank of E, s is the rank of its 2-Selmer group and v is a list of independant points in E(Q)/2E(Q). If help is a vector of nontrivial points on E, the result might be faster. This function might be used in conjunction with elltors2(E). See also ?default_ellQ");
addhelp(ellhalf,
"ellhalf(E,P): returns the vector of all points Q on the elliptic curve E such that 2Q = P");
addhelp(ellredgen,
@@ -2143,7 +2143,7 @@ if( DEBUGLEVEL_ell >= 3, print(" end of ell2descent_viaisog"));
\\ others
addhelp(default_ellQ,
- "default_ellQ(DEBUGLEVEL_ell, LIM1, LIM3, LIMTRIV, ELLREDGENFLAG, COMPLETE, MAXPROB, LIMBIGPRIME): set the value of the global variables used for ellrank() and other related functions. DEBUGLEVEL_ell: 0-5: choose the quantity of information printed during the computation (default=0: print nothing); LIM1 (resp LIM3): search limit for easy (resp hard) points on quartics; LIMTRIV: search limit for trivial points on elliptic curves; ELLREDGENFLAG: if != 0, try to reduce the generators at the end; COMPLETE: if != 0 and full 2-torsion, use complete 2-descent, otherwise via 2-isogeny; MAXPROB, LIMBIGPRIME: technical.");
+ "default_ellQ(DEBUGLEVEL_ell, LIM1, LIM3, LIMTRIV, ELLREDGENFLAG, COMPLETE, MAXPROB, LIMBIGPRIME): set the value of the global variables used for ellQ_ellrank() and other related functions. DEBUGLEVEL_ell: 0-5: choose the quantity of information printed during the computation (default=0: print nothing); LIM1 (resp LIM3): search limit for easy (resp hard) points on quartics; LIMTRIV: search limit for trivial points on elliptic curves; ELLREDGENFLAG: if != 0, try to reduce the generators at the end; COMPLETE: if != 0 and full 2-torsion, use complete 2-descent, otherwise via 2-isogeny; MAXPROB, LIMBIGPRIME: technical.");
/* addhelp(DEBUGLEVEL_ell,
"DEBUGLEVEL_ell: Choose a higher value of this global variable to have more details of the computations printed during the 2-descent algorithm. 0 = don't print anything; 1 = (default) just print the result; 2 = print more details including the Selmer group and the nontrivial quartics.");
*/
diff --git a/src/sage/ext_data/pari/simon/qfsolve.gp b/src/sage/ext_data/pari/simon/qfsolve.gp
index 501fb50828..2107288c1d 100644
--- a/src/sage/ext_data/pari/simon/qfsolve.gp
+++ b/src/sage/ext_data/pari/simon/qfsolve.gp
@@ -434,146 +434,6 @@ my(cc);
return([U3~*G3*U3,red[2]*U1*U2*U3]);
}
-\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
-\\ QUADRATIC FORMS MINIMIZATION \\
-\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
-
-\\ Minimization of the quadratic form G, with nonzero determinant.
-\\ of dimension n>=2.
-\\ G must by symmetric and have integral coefficients.
-\\ Returns [G',U,factd] with U in GLn(Q) such that G'=U~*G*U*constant
-\\ is integral and has minimal determinant.
-\\ In dimension 3 or 4, may return a prime p
-\\ if the reduction at p is impossible because of the local non solvability.
-\\ If given, factdetG must be equal to factor(abs(det(G))).
-{qfminimize(G,factdetG) =
-my(factd,U,Ker,Ker2,sol,aux,di);
-my(p);
-my(n,lf,i,vp,dimKer,dimKer2,m);
-
- n = length(G);
- factd = matrix(0,2);
- if( !factdetG, factdetG = factor(matdet(G)));
-
- lf = length(factdetG[,1]);
- i = 1; U = matid(n);
-
- while(i <= lf,
- vp = factdetG[i,2];
- if( vp == 0, i++; next);
- p = factdetG[i,1];
- if( p == -1, i++; next);
-if( DEBUGLEVEL_qfsolve >= 4, print(" p = ",p,"^",vp));
-
-\\ The case vp = 1 can be minimized only if n is odd.
- if( vp == 1 && n%2 == 0,
- factd = concat(factd~, Mat([p,1])~)~;
- i++; next
- );
- Ker = kermodp(G,p); dimKer = Ker[1]; Ker = Ker[2];
-
-\\ Rem: we must have dimKer <= vp
-if( DEBUGLEVEL_qfsolve >= 4, print(" dimKer = ",dimKer));
-\\ trivial case: dimKer = n
- if( dimKer == n,
-if( DEBUGLEVEL_qfsolve >= 4, print(" case 0: dimKer = n"));
- G /= p;
- factdetG[i,2] -= n;
- next
- );
- G = Ker~*G*Ker;
- U = U*Ker;
-
-\\ 1st case: dimKer < vp
-\\ then the kernel mod p contains a kernel mod p^2
- if( dimKer < vp,
-if( DEBUGLEVEL_qfsolve >= 4, print(" case 1: dimker < vp"));
- if( dimKer == 1,
-\\ G[,1] /= p; G[1,] /= p;
- G[,1] /= p; G[1,] = G[1,]/p;
- U[,1] /= p;
- factdetG[i,2] -= 2
- ,
- Ker2 = kermodp(matrix(dimKer,dimKer,j,k,G[j,k]/p),p);
- dimKer2 = Ker2[1]; Ker2 = Ker2[2];
- for( j = 1, dimKer2, Ker2[,j] /= p);
- Ker2 = matdiagonalblock([Ker2,matid(n-dimKer)]);
- G = Ker2~*G*Ker2;
- U = U*Ker2;
- factdetG[i,2] -= 2*dimKer2
-);
-
-if( DEBUGLEVEL_qfsolve >= 4, print(" end of case 1"));
- next
- );
-
-\\ Now, we have vp = dimKer
-\\ 2nd case: the dimension of the kernel is >=2
-\\ and contains an element of norm 0 mod p^2
-
-\\ search for an element of norm p^2... in the kernel
- if( dimKer > 2 ||
- (dimKer == 2 && issquare( di = Mod((G[1,2]^2-G[1,1]*G[2,2])/p^2,p))),
- if( dimKer > 2,
-if( DEBUGLEVEL_qfsolve >= 4, print(" case 2.1"));
- dimKer = 3;
- sol = qfsolvemodp(matrix(3,3,j,k,G[j,k]/p),p)
- ,
-if( DEBUGLEVEL_qfsolve >= 4, print(" case 2.2"));
- if( G[1,1]%p^2 == 0,
- sol = [1,0]~
- , sol = [-G[1,2]/p+sqrt(di),Mod(G[1,1]/p,p)]~
- )
- );
- sol = centerlift(sol);
- sol /= content(sol);
-if( DEBUGLEVEL_qfsolve >= 4, print(" sol = ",sol));
- Ker = vectorv(n, j, if( j<= dimKer, sol[j], 0)); \\ fill with 0's
- Ker = completebasis(Ker,1);
- Ker[,n] /= p;
- G = Ker~*G*Ker;
- U = U*Ker;
- factdetG[i,2] -= 2;
-if( DEBUGLEVEL_qfsolve >= 4, print(" end of case 2"));
- next
- );
-
-\\ Now, we have vp = dimKer <= 2
-\\ and the kernel contains no vector with norm p^2...
-
-\\ In some cases, exchanging the kernel and the image
-\\ makes the minimization easy.
-
- m = (n-1)\2-1;
- if( ( vp == 1 && issquare(Mod(-(-1)^m*matdet(G)/G[1,1],p)))
- || ( vp == 2 && n%2 == 1 && n >= 5)
- || ( vp == 2 && n%2 == 0 && !issquare(Mod((-1)^m*matdet(G)/p^2,p)))
- ,
-if( DEBUGLEVEL_qfsolve >= 4, print(" case 3"));
- Ker = matid(n);
- for( j = dimKer+1, n, Ker[j,j] = p);
- G = Ker~*G*Ker/p;
- U = U*Ker;
- factdetG[i,2] -= 2*dimKer-n;
-if( DEBUGLEVEL_qfsolve >= 4, print(" end of case 3"));
- next
- );
-
-\\ Minimization was not possible se far.
-\\ If n == 3 or 4, this proves the local non-solubility at p.
- if( n == 3 || n == 4,
-if( DEBUGLEVEL_qfsolve >= 1, print(" no local solution at ",p));
- return(p));
-
-if( DEBUGLEVEL_qfsolve >= 4, print(" prime ",p," finished"));
- factd = concat(factd~,Mat([p,vp])~)~;
- i++
- );
-\\ apply LLL to avoid coefficients explosion
- aux = qflll(U/content(U));
-return([aux~*G*aux,U*aux,factd]);
-}
-
\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ CLASS GROUP COMPUTATIONS \\
\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
diff --git a/src/sage/geometry/polyhedron/backend_field.py b/src/sage/geometry/polyhedron/backend_field.py
index 6b921d23a6..2f32c58b1e 100644
--- a/src/sage/geometry/polyhedron/backend_field.py
+++ b/src/sage/geometry/polyhedron/backend_field.py
@@ -265,7 +265,7 @@ class Polyhedron_field(Polyhedron_base):
An inequality (-0.1419794359520263?, -1.698172434277148?) x + 1.200789243901438? >= 0,
An inequality (0.3001973109753594?, 0.600394621950719?) x - 0.4245431085692869? >= 0)
sage: p.Vrepresentation() # optional - sage.rings.number_field
- (A vertex at (0.?e-15, 0.707106781186548?),
+ (A vertex at (0.?e-16, 0.7071067811865475?),
A vertex at (1.414213562373095?, 0),
A vertex at (4.000000000000000?, 0.372677996249965?))
"""
@@ -308,7 +308,7 @@ class Polyhedron_field(Polyhedron_base):
An inequality (-0.1419794359520263?, -1.698172434277148?) x + 1.200789243901438? >= 0,
An inequality (0.3001973109753594?, 0.600394621950719?) x - 0.4245431085692869? >= 0)
sage: p.Vrepresentation() # optional - sage.rings.number_field
- (A vertex at (0.?e-15, 0.707106781186548?),
+ (A vertex at (0.?e-16, 0.7071067811865475?),
A vertex at (1.414213562373095?, 0),
A vertex at (4.000000000000000?, 0.372677996249965?))
"""
diff --git a/src/sage/geometry/polyhedron/backend_normaliz.py b/src/sage/geometry/polyhedron/backend_normaliz.py
index 86b89632a5..ca8a43b248 100644
--- a/src/sage/geometry/polyhedron/backend_normaliz.py
+++ b/src/sage/geometry/polyhedron/backend_normaliz.py
@@ -53,7 +53,7 @@ def _number_field_elements_from_algebraics_list_of_lists_of_lists(listss, **kwds
1.732050807568878?
sage: from sage.geometry.polyhedron.backend_normaliz import _number_field_elements_from_algebraics_list_of_lists_of_lists
sage: K, results, hom = _number_field_elements_from_algebraics_list_of_lists_of_lists([[[rt2], [1]], [[rt3]], [[1], []]]); results # optional - sage.rings.number_field
- [[[-a^3 + 3*a], [1]], [[-a^2 + 2]], [[1], []]]
+ [[[-a^3 + 3*a], [1]], [[a^2 - 2]], [[1], []]]
"""
from sage.rings.qqbar import number_field_elements_from_algebraics
numbers = []
diff --git a/src/sage/groups/matrix_gps/isometries.py b/src/sage/groups/matrix_gps/isometries.py
index f9111a2c92..cca45e7175 100644
--- a/src/sage/groups/matrix_gps/isometries.py
+++ b/src/sage/groups/matrix_gps/isometries.py
@@ -11,11 +11,11 @@ EXAMPLES::
sage: L = IntegralLattice("D4")
sage: O = L.orthogonal_group()
sage: O
- Group of isometries with 5 generators (
- [-1 0 0 0] [0 0 0 1] [-1 -1 -1 -1] [ 1 1 0 0] [ 1 0 0 0]
- [ 0 -1 0 0] [0 1 0 0] [ 0 0 1 0] [ 0 0 1 0] [-1 -1 -1 -1]
- [ 0 0 -1 0] [0 0 1 0] [ 0 1 0 1] [ 0 1 0 1] [ 0 0 1 0]
- [ 0 0 0 -1], [1 0 0 0], [ 0 -1 -1 0], [ 0 -1 -1 0], [ 0 0 0 1]
+ Group of isometries with 3 generators (
+ [0 0 0 1] [ 1 1 0 0] [ 1 0 0 0]
+ [0 1 0 0] [ 0 0 1 0] [-1 -1 -1 -1]
+ [0 0 1 0] [ 0 1 0 1] [ 0 0 1 0]
+ [1 0 0 0], [ 0 -1 -1 0], [ 0 0 0 1]
)
Basic functionality is provided by GAP::
diff --git a/src/sage/interfaces/genus2reduction.py b/src/sage/interfaces/genus2reduction.py
index 56ae04b235..7a4794daf2 100644
--- a/src/sage/interfaces/genus2reduction.py
+++ b/src/sage/interfaces/genus2reduction.py
@@ -143,31 +143,31 @@ class ReductionData(SageObject):
sur un corps de valuation discrète", Trans. AMS 348 (1996),
4577-4610, Section 7.2, Proposition 4).
"""
- def __init__(self, pari_result, P, Q, minimal_equation, minimal_disc,
- local_data, conductor, prime_to_2_conductor_only):
+ def __init__(self, pari_result, P, Q, Pmin, Qmin, minimal_disc,
+ local_data, conductor):
self.pari_result = pari_result
self.P = P
self.Q = Q
- self.minimal_equation = minimal_equation
+ self.Pmin = Pmin
+ self.Qmin = Qmin
self.minimal_disc = minimal_disc
self.local_data = local_data
self.conductor = conductor
- self.prime_to_2_conductor_only = prime_to_2_conductor_only
def _repr_(self):
- if self.prime_to_2_conductor_only:
- ex = ' (away from 2)'
- else:
- ex = ''
if self.Q == 0:
yterm = ''
else:
yterm = '+ (%s)*y '%self.Q
+
s = 'Reduction data about this proper smooth genus 2 curve:\n'
s += '\ty^2 %s= %s\n'%(yterm, self.P)
- s += 'A Minimal Equation (away from 2):\n\ty^2 = %s\n'%self.minimal_equation
- s += 'Minimal Discriminant (away from 2): %s\n'%self.minimal_disc
- s += 'Conductor%s: %s\n'%(ex, self.conductor)
+ if self.Qmin:
+ s += 'A Minimal Equation:\n\ty^2 + (%s)y = %s\n'%(self.Qmin, self.Pmin)
+ else:
+ s += 'A Minimal Equation:\n\ty^2 = %s\n'%self.Pmin
+ s += 'Minimal Discriminant: %s\n'%self.minimal_disc
+ s += 'Conductor: %s\n'%self.conductor
s += 'Local Data:\n%s'%self._local_data_str()
return s
@@ -242,17 +242,7 @@ class Genus2reduction(SageObject):
sage: factor(R.conductor)
5^4 * 2267
- This means that only the odd part of the conductor is known.
-
- ::
-
- sage: R.prime_to_2_conductor_only
- True
-
- The discriminant is always minimal away from 2, but possibly not at
- 2.
-
- ::
+ The discriminant is always minimal::
sage: factor(R.minimal_disc)
2^3 * 5^5 * 2267
@@ -264,10 +254,10 @@ class Genus2reduction(SageObject):
sage: R
Reduction data about this proper smooth genus 2 curve:
y^2 + (x^3 - 2*x^2 - 2*x + 1)*y = -5*x^5
- A Minimal Equation (away from 2):
- y^2 = x^6 - 240*x^4 - 2550*x^3 - 11400*x^2 - 24100*x - 19855
- Minimal Discriminant (away from 2): 56675000
- Conductor (away from 2): 1416875
+ A Minimal Equation:
+ y^2 ...
+ Minimal Discriminant: 56675000
+ Conductor: 1416875
Local Data:
p=2
(potential) stable reduction: (II), j=1
@@ -293,10 +283,10 @@ class Genus2reduction(SageObject):
sage: genus2reduction(0, x^6 + 3*x^3 + 63)
Reduction data about this proper smooth genus 2 curve:
y^2 = x^6 + 3*x^3 + 63
- A Minimal Equation (away from 2):
- y^2 = x^6 + 3*x^3 + 63
- Minimal Discriminant (away from 2): 10628388316852992
- Conductor (away from 2): 2893401
+ A Minimal Equation:
+ y^2 ...
+ Minimal Discriminant: -10628388316852992
+ Conductor: 2893401
Local Data:
p=2
(potential) stable reduction: (V), j1+j2=0, j1*j2=0
@@ -327,9 +317,9 @@ class Genus2reduction(SageObject):
sage: genus2reduction(x^3-x^2-1, x^2 - x)
Reduction data about this proper smooth genus 2 curve:
y^2 + (x^3 - x^2 - 1)*y = x^2 - x
- A Minimal Equation (away from 2):
- y^2 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561
- Minimal Discriminant (away from 2): 169
+ A Minimal Equation:
+ y^2 ...
+ Minimal Discriminant: -169
Conductor: 169
Local Data:
p=13
@@ -370,10 +360,10 @@ class Genus2reduction(SageObject):
sage: genus2reduction(x^3 - 2*x^2 - 2*x + 1, -5*x^5)
Reduction data about this proper smooth genus 2 curve:
y^2 + (x^3 - 2*x^2 - 2*x + 1)*y = -5*x^5
- A Minimal Equation (away from 2):
- y^2 = x^6 - 240*x^4 - 2550*x^3 - 11400*x^2 - 24100*x - 19855
- Minimal Discriminant (away from 2): 56675000
- Conductor (away from 2): 1416875
+ A Minimal Equation:
+ y^2 ...
+ Minimal Discriminant: 56675000
+ Conductor: 1416875
Local Data:
p=2
(potential) stable reduction: (II), j=1
@@ -389,9 +379,9 @@ class Genus2reduction(SageObject):
sage: genus2reduction(x^2 + 1, -5*x^5)
Reduction data about this proper smooth genus 2 curve:
y^2 + (x^2 + 1)*y = -5*x^5
- A Minimal Equation (away from 2):
- y^2 = -20*x^5 + x^4 + 2*x^2 + 1
- Minimal Discriminant (away from 2): 48838125
+ A Minimal Equation:
+ y^2 ...
+ Minimal Discriminant: 48838125
Conductor: 32025
Local Data:
p=3
@@ -412,9 +402,9 @@ class Genus2reduction(SageObject):
sage: genus2reduction(x^3 + x^2 + x,-2*x^5 + 3*x^4 - x^3 - x^2 - 6*x - 2)
Reduction data about this proper smooth genus 2 curve:
y^2 + (x^3 + x^2 + x)*y = -2*x^5 + 3*x^4 - x^3 - x^2 - 6*x - 2
- A Minimal Equation (away from 2):
- y^2 = x^6 + 18*x^3 + 36*x^2 - 27
- Minimal Discriminant (away from 2): 1520984142
+ A Minimal Equation:
+ y^2 ...
+ Minimal Discriminant: 1520984142
Conductor: 954
Local Data:
p=2
@@ -436,18 +426,10 @@ class Genus2reduction(SageObject):
raise ValueError("Q (=%s) must have degree at most 3" % Q)
res = pari.genus2red([P, Q])
-
conductor = ZZ(res[0])
- minimal_equation = R(res[2])
-
- minimal_disc = QQ(res[2].poldisc()).abs()
- if minimal_equation.degree() == 5:
- minimal_disc *= minimal_equation[5]**2
- # Multiply with suitable power of 2 of the form 2^(2*(d-1) - 12)
- b = 2 * (minimal_equation.degree() - 1)
- k = QQ((12 - minimal_disc.valuation(2), b)).ceil()
- minimal_disc >>= 12 - b*k
- minimal_disc = ZZ(minimal_disc)
+ Pmin = R(res[2][0])
+ Qmin = R(res[2][1])
+ minimal_disc = ZZ(pari.hyperelldisc(res[2]))
local_data = {}
for red in res[3]:
@@ -468,9 +450,7 @@ class Genus2reduction(SageObject):
local_data[p] = data
- prime_to_2_conductor_only = (-1 in res[1].component(2))
- return ReductionData(res, P, Q, minimal_equation, minimal_disc, local_data,
- conductor, prime_to_2_conductor_only)
+ return ReductionData(res, P, Q, Pmin, Qmin, minimal_disc, local_data, conductor)
def __reduce__(self):
return _reduce_load_genus2reduction, tuple([])
diff --git a/src/sage/lfunctions/dokchitser.py b/src/sage/lfunctions/dokchitser.py
index fec450d7bc..236402c293 100644
--- a/src/sage/lfunctions/dokchitser.py
+++ b/src/sage/lfunctions/dokchitser.py
@@ -337,6 +337,7 @@ class Dokchitser(SageObject):
# After init_coeffs is called, future calls to this method should
# return the full output for further parsing
raise RuntimeError("unable to create L-series, due to precision or other limits in PARI")
+ t = t.replace(" *** _^_: Warning: normalizing a series with 0 leading term.\n", "")
return t
def __check_init(self):
diff --git a/src/sage/lfunctions/pari.py b/src/sage/lfunctions/pari.py
index d2b20f1891..6c31efe239 100644
--- a/src/sage/lfunctions/pari.py
+++ b/src/sage/lfunctions/pari.py
@@ -339,7 +339,7 @@ def lfun_eta_quotient(scalings, exponents):
0.0374412812685155
sage: lfun_eta_quotient([6],[4])
- [[Vecsmall([7]), [Vecsmall([6]), Vecsmall([4])]], 0, [0, 1], 2, 36, 1]
+ [[Vecsmall([7]), [Vecsmall([6]), Vecsmall([4]), 0]], 0, [0, 1], 2, 36, 1]
sage: lfun_eta_quotient([2,1,4], [5,-2,-2])
Traceback (most recent call last):
diff --git a/src/sage/libs/pari/tests.py b/src/sage/libs/pari/tests.py
index e5a2aa2517..0efcb15de0 100644
--- a/src/sage/libs/pari/tests.py
+++ b/src/sage/libs/pari/tests.py
@@ -356,7 +356,7 @@ Constructors::
[2, 4]~*x + [1, 3]~
sage: pari(3).Qfb(7, 1)
- Qfb(3, 7, 1, 0.E-19)
+ Qfb(3, 7, 1)
sage: pari(3).Qfb(7, 2)
Traceback (most recent call last):
...
@@ -512,7 +512,7 @@ Basic functions::
sage: pari('sqrt(-2)').frac()
Traceback (most recent call last):
...
- PariError: incorrect type in gfloor (t_COMPLEX)
+ PariError: incorrect type in gfrac (t_COMPLEX)
sage: pari('1+2*I').imag()
2
diff --git a/src/sage/modular/cusps_nf.py b/src/sage/modular/cusps_nf.py
index 25d93cac92..157ebabe29 100644
--- a/src/sage/modular/cusps_nf.py
+++ b/src/sage/modular/cusps_nf.py
@@ -1220,7 +1220,7 @@ def units_mod_ideal(I):
sage: I = k.ideal(5, a + 1)
sage: units_mod_ideal(I)
[1,
- 2*a^2 + 4*a - 1,
+ -2*a^2 - 4*a + 1,
...]
::
diff --git a/src/sage/modular/modform_hecketriangle/hecke_triangle_group_element.py b/src/sage/modular/modform_hecketriangle/hecke_triangle_group_element.py
index a881336596..090d1bfaf0 100644
--- a/src/sage/modular/modform_hecketriangle/hecke_triangle_group_element.py
+++ b/src/sage/modular/modform_hecketriangle/hecke_triangle_group_element.py
@@ -43,7 +43,7 @@ def coerce_AA(p):
sage: AA(p)._exact_field()
Number Field in a with defining polynomial y^8 ... with a in ...
sage: coerce_AA(p)._exact_field()
- Number Field in a with defining polynomial y^4 - 1910*y^2 - 3924*y + 681058 with a in 39.710518724...?
+ Number Field in a with defining polynomial y^4 - 1910*y^2 - 3924*y + 681058 with a in ...?
"""
el = AA(p)
el.simplify()
diff --git a/src/sage/modular/modsym/p1list_nf.py b/src/sage/modular/modsym/p1list_nf.py
index 222caacca8..f9d969732c 100644
--- a/src/sage/modular/modsym/p1list_nf.py
+++ b/src/sage/modular/modsym/p1list_nf.py
@@ -58,7 +58,7 @@ Lift an MSymbol to a matrix in `SL(2, R)`:
sage: alpha = MSymbol(N, a + 2, 3*a^2)
sage: alpha.lift_to_sl2_Ok()
- [-3*a^2 + a + 12, 25*a^2 - 50*a + 100, a + 2, a^2 - 3*a + 3]
+ [-1, 4*a^2 - 13*a + 23, a + 2, 5*a^2 + 3*a - 3]
sage: Ok = k.ring_of_integers()
sage: M = Matrix(Ok, 2, alpha.lift_to_sl2_Ok())
sage: det(M)
@@ -945,11 +945,11 @@ class P1NFList(SageObject):
sage: N = k.ideal(5, a + 1)
sage: P = P1NFList(N)
sage: u = k.unit_group().gens_values(); u
- [-1, 2*a^2 + 4*a - 1]
+ [-1, -2*a^2 - 4*a + 1]
sage: P.apply_J_epsilon(4, -1)
2
sage: P.apply_J_epsilon(4, u[0], u[1])
- 1
+ 5
::
diff --git a/src/sage/modules/free_quadratic_module_integer_symmetric.py b/src/sage/modules/free_quadratic_module_integer_symmetric.py
index a206f0c721..aeb19ab669 100644
--- a/src/sage/modules/free_quadratic_module_integer_symmetric.py
+++ b/src/sage/modules/free_quadratic_module_integer_symmetric.py
@@ -1168,11 +1168,11 @@ class FreeQuadraticModule_integer_symmetric(FreeQuadraticModule_submodule_with_b
sage: A4 = IntegralLattice("A4")
sage: Aut = A4.orthogonal_group()
sage: Aut
- Group of isometries with 5 generators (
- [-1 0 0 0] [0 0 0 1] [-1 -1 -1 0] [ 1 0 0 0] [ 1 0 0 0]
- [ 0 -1 0 0] [0 0 1 0] [ 0 0 0 -1] [-1 -1 -1 -1] [ 0 1 0 0]
- [ 0 0 -1 0] [0 1 0 0] [ 0 0 1 1] [ 0 0 0 1] [ 0 0 1 1]
- [ 0 0 0 -1], [1 0 0 0], [ 0 1 0 0], [ 0 0 1 0], [ 0 0 0 -1]
+ Group of isometries with 4 generators (
+ [0 0 0 1] [-1 -1 -1 0] [ 1 0 0 0] [ 1 0 0 0]
+ [0 0 1 0] [ 0 0 0 -1] [-1 -1 -1 -1] [ 0 1 0 0]
+ [0 1 0 0] [ 0 0 1 1] [ 0 0 0 1] [ 0 0 1 1]
+ [1 0 0 0], [ 0 1 0 0], [ 0 0 1 0], [ 0 0 0 -1]
)
The group acts from the right on the lattice and its discriminant group::
@@ -1180,19 +1180,19 @@ class FreeQuadraticModule_integer_symmetric(FreeQuadraticModule_submodule_with_b
sage: x = A4.an_element()
sage: g = Aut.an_element()
sage: g
- [ 1 1 1 0]
- [ 0 0 -1 0]
- [ 0 0 1 1]
- [ 0 -1 -1 -1]
+ [-1 -1 -1 0]
+ [ 0 0 1 0]
+ [ 0 0 -1 -1]
+ [ 0 1 1 1]
sage: x*g
- (1, 1, 1, 0)
+ (-1, -1, -1, 0)
sage: (x*g).parent()==A4
True
sage: (g*x).parent()
Vector space of dimension 4 over Rational Field
sage: y = A4.discriminant_group().an_element()
sage: y*g
- (1)
+ (4)
If the group is finite we can compute the usual things::
@@ -1208,10 +1208,10 @@ class FreeQuadraticModule_integer_symmetric(FreeQuadraticModule_submodule_with_b
sage: A2 = IntegralLattice(matrix.identity(3),Matrix(ZZ,2,3,[1,-1,0,0,1,-1]))
sage: A2.orthogonal_group()
- Group of isometries with 3 generators (
- [-1/3 2/3 2/3] [ 2/3 2/3 -1/3] [1 0 0]
- [ 2/3 -1/3 2/3] [ 2/3 -1/3 2/3] [0 0 1]
- [ 2/3 2/3 -1/3], [-1/3 2/3 2/3], [0 1 0]
+ Group of isometries with 2 generators (
+ [ 2/3 2/3 -1/3] [1 0 0]
+ [ 2/3 -1/3 2/3] [0 0 1]
+ [-1/3 2/3 2/3], [0 1 0]
)
It can be negative definite as well::
diff --git a/src/sage/quadratic_forms/binary_qf.py b/src/sage/quadratic_forms/binary_qf.py
index cfa3ada73e..5ac823bc6c 100755
--- a/src/sage/quadratic_forms/binary_qf.py
+++ b/src/sage/quadratic_forms/binary_qf.py
@@ -141,7 +141,7 @@ class BinaryQF(SageObject):
and a.degree() == 2 and a.parent().ngens() == 2):
x, y = a.parent().gens()
a, b, c = [a.monomial_coefficient(mon) for mon in [x**2, x*y, y**2]]
- elif isinstance(a, pari_gen) and a.type() in ('t_QFI', 't_QFR'):
+ elif isinstance(a, pari_gen) and a.type() in ('t_QFI', 't_QFR', 't_QFB'):
# a has 3 or 4 components
a, b, c = a[0], a[1], a[2]
try:
diff --git a/src/sage/quadratic_forms/genera/genus.py b/src/sage/quadratic_forms/genera/genus.py
index 8290b6c4fa..0fc43f33c6 100644
--- a/src/sage/quadratic_forms/genera/genus.py
+++ b/src/sage/quadratic_forms/genera/genus.py
@@ -3088,8 +3088,8 @@ class GenusSymbol_global_ring():
sage: G = Genus(matrix(ZZ, 3, [6,3,0, 3,6,0, 0,0,2]))
sage: G.representatives()
(
- [2 0 0] [ 2 -1 0]
- [0 6 3] [-1 2 0]
+ [2 0 0] [ 2 1 0]
+ [0 6 3] [ 1 2 0]
[0 3 6], [ 0 0 18]
)
diff --git a/src/sage/quadratic_forms/qfsolve.py b/src/sage/quadratic_forms/qfsolve.py
index ddde95e04f..d5e15d9f83 100644
--- a/src/sage/quadratic_forms/qfsolve.py
+++ b/src/sage/quadratic_forms/qfsolve.py
@@ -70,7 +70,7 @@ def qfsolve(G):
sage: M = Matrix(QQ, [[3, 0, 0, 0], [0, 5, 0, 0], [0, 0, -7, 0], [0, 0, 0, -11]])
sage: qfsolve(M)
- (3, -4, -3, -2)
+ (3, 4, -3, -2)
"""
ret = G.__pari__().qfsolve()
if ret.type() == 't_COL':
diff --git a/src/sage/quadratic_forms/quadratic_form__automorphisms.py b/src/sage/quadratic_forms/quadratic_form__automorphisms.py
index c36c667e3b..3d72cf3be1 100644
--- a/src/sage/quadratic_forms/quadratic_form__automorphisms.py
+++ b/src/sage/quadratic_forms/quadratic_form__automorphisms.py
@@ -300,9 +300,9 @@ def automorphism_group(self):
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.automorphism_group()
Matrix group over Rational Field with 3 generators (
- [-1 0 0] [0 0 1] [ 0 0 1]
- [ 0 -1 0] [0 1 0] [-1 0 0]
- [ 0 0 -1], [1 0 0], [ 0 1 0]
+ [ 0 0 1] [1 0 0] [ 1 0 0]
+ [-1 0 0] [0 0 1] [ 0 -1 0]
+ [ 0 1 0], [0 1 0], [ 0 0 1]
)
::
diff --git a/src/sage/rings/finite_rings/finite_field_prime_modn.py b/src/sage/rings/finite_rings/finite_field_prime_modn.py
index 9129ecb56a..d5a4cb8f22 100644
--- a/src/sage/rings/finite_rings/finite_field_prime_modn.py
+++ b/src/sage/rings/finite_rings/finite_field_prime_modn.py
@@ -111,7 +111,7 @@ class FiniteField_prime_modn(FiniteField_generic, integer_mod_ring.IntegerModRin
sage: RF13 = K.residue_field(pp)
sage: RF13.hom([GF(13)(1)])
Ring morphism:
- From: Residue field of Fractional ideal (w + 18)
+ From: Residue field of Fractional ideal (-w - 18)
To: Finite Field of size 13
Defn: 1 |--> 1
diff --git a/src/sage/rings/finite_rings/residue_field.pyx b/src/sage/rings/finite_rings/residue_field.pyx
index 7596f2a302..1e1869f1b1 100644
--- a/src/sage/rings/finite_rings/residue_field.pyx
+++ b/src/sage/rings/finite_rings/residue_field.pyx
@@ -20,13 +20,13 @@ monogenic (i.e., 2 is an essential discriminant divisor)::
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: F = K.factor(2); F
- (Fractional ideal (1/2*a^2 - 1/2*a + 1)) * (Fractional ideal (-a^2 + 2*a - 3)) * (Fractional ideal (-3/2*a^2 + 5/2*a - 4))
+ (Fractional ideal (-1/2*a^2 + 1/2*a - 1)) * (Fractional ideal (-a^2 + 2*a - 3)) * (Fractional ideal (3/2*a^2 - 5/2*a + 4))
sage: F[0][0].residue_field()
- Residue field of Fractional ideal (1/2*a^2 - 1/2*a + 1)
+ Residue field of Fractional ideal (-1/2*a^2 + 1/2*a - 1)
sage: F[1][0].residue_field()
Residue field of Fractional ideal (-a^2 + 2*a - 3)
sage: F[2][0].residue_field()
- Residue field of Fractional ideal (-3/2*a^2 + 5/2*a - 4)
+ Residue field of Fractional ideal (3/2*a^2 - 5/2*a + 4)
We can also form residue fields from `\ZZ`::
@@ -258,9 +258,9 @@ class ResidueFieldFactory(UniqueFactory):
the index of ``ZZ[a]`` in the maximal order for all ``a``::
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8); P = K.ideal(2).factor()[0][0]; P
- Fractional ideal (1/2*a^2 - 1/2*a + 1)
+ Fractional ideal (-1/2*a^2 + 1/2*a - 1)
sage: F = K.residue_field(P); F
- Residue field of Fractional ideal (1/2*a^2 - 1/2*a + 1)
+ Residue field of Fractional ideal (-1/2*a^2 + 1/2*a - 1)
sage: F(a)
0
sage: B = K.maximal_order().basis(); B
@@ -270,7 +270,7 @@ class ResidueFieldFactory(UniqueFactory):
sage: F(B[2])
0
sage: F
- Residue field of Fractional ideal (1/2*a^2 - 1/2*a + 1)
+ Residue field of Fractional ideal (-1/2*a^2 + 1/2*a - 1)
sage: F.degree()
1
@@ -730,15 +730,15 @@ class ResidueField_generic(Field):
EXAMPLES::
sage: I = QQ[3^(1/3)].factor(5)[1][0]; I
- Fractional ideal (-a + 2)
+ Fractional ideal (a - 2)
sage: k = I.residue_field(); k
- Residue field of Fractional ideal (-a + 2)
+ Residue field of Fractional ideal (a - 2)
sage: f = k.lift_map(); f
Lifting map:
- From: Residue field of Fractional ideal (-a + 2)
+ From: Residue field of Fractional ideal (a - 2)
To: Maximal Order in Number Field in a with defining polynomial x^3 - 3 with a = 1.442249570307409?
sage: f.domain()
- Residue field of Fractional ideal (-a + 2)
+ Residue field of Fractional ideal (a - 2)
sage: f.codomain()
Maximal Order in Number Field in a with defining polynomial x^3 - 3 with a = 1.442249570307409?
sage: f(k.0)
@@ -768,7 +768,7 @@ class ResidueField_generic(Field):
sage: K.<a> = NumberField(x^3-11)
sage: F = K.ideal(37).factor(); F
- (Fractional ideal (37, a + 9)) * (Fractional ideal (37, a + 12)) * (Fractional ideal (2*a - 5))
+ (Fractional ideal (37, a + 9)) * (Fractional ideal (37, a + 12)) * (Fractional ideal (-2*a + 5))
sage: k = K.residue_field(F[0][0])
sage: l = K.residue_field(F[1][0])
sage: k == l
@@ -846,7 +846,7 @@ cdef class ReductionMap(Map):
sage: F.reduction_map()
Partially defined reduction map:
From: Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8
- To: Residue field of Fractional ideal (1/2*a^2 - 1/2*a + 1)
+ To: Residue field of Fractional ideal (-1/2*a^2 + 1/2*a - 1)
sage: K.<theta_5> = CyclotomicField(5)
sage: F = K.factor(7)[0][0].residue_field()
diff --git a/src/sage/rings/number_field/S_unit_solver.py b/src/sage/rings/number_field/S_unit_solver.py
index e99dff850f..759cbfb334 100644
--- a/src/sage/rings/number_field/S_unit_solver.py
+++ b/src/sage/rings/number_field/S_unit_solver.py
@@ -1781,20 +1781,20 @@ def sieve_ordering(SUK, q):
sage: SUK = K.S_unit_group(S=3)
sage: sieve_data = list(sieve_ordering(SUK, 19))
sage: sieve_data[0]
- (Fractional ideal (xi - 3),
- Fractional ideal (-2*xi^2 + 3),
+ (Fractional ideal (-2*xi^2 + 3),
+ Fractional ideal (-xi + 3),
Fractional ideal (2*xi + 1))
sage: sieve_data[1]
- (Residue field of Fractional ideal (xi - 3),
- Residue field of Fractional ideal (-2*xi^2 + 3),
+ (Residue field of Fractional ideal (-2*xi^2 + 3),
+ Residue field of Fractional ideal (-xi + 3),
Residue field of Fractional ideal (2*xi + 1))
sage: sieve_data[2]
- ([18, 7, 16, 4], [18, 9, 12, 8], [18, 3, 10, 10])
+ ([18, 12, 16, 8], [18, 16, 10, 4], [18, 10, 12, 10])
sage: sieve_data[3]
- (486, 648, 11664)
+ (648, 2916, 3888)
"""
K = SUK.number_field()
diff --git a/src/sage/rings/number_field/bdd_height.py b/src/sage/rings/number_field/bdd_height.py
index beb047ae02..b7c8c33d0b 100644
--- a/src/sage/rings/number_field/bdd_height.py
+++ b/src/sage/rings/number_field/bdd_height.py
@@ -248,7 +248,7 @@ def bdd_norm_pr_ideal_gens(K, norm_list):
sage: from sage.rings.number_field.bdd_height import bdd_norm_pr_ideal_gens
sage: K.<g> = QuadraticField(123)
sage: bdd_norm_pr_ideal_gens(K, range(5))
- {0: [0], 1: [1], 2: [-g - 11], 3: [], 4: [2]}
+ {0: [0], 1: [1], 2: [g + 11], 3: [], 4: [2]}
::
diff --git a/src/sage/rings/number_field/class_group.py b/src/sage/rings/number_field/class_group.py
index 018ff5f5c6..73c0462cd1 100644
--- a/src/sage/rings/number_field/class_group.py
+++ b/src/sage/rings/number_field/class_group.py
@@ -221,11 +221,11 @@ class FractionalIdealClass(AbelianGroupWithValuesElement):
Class group of order 76 with structure C38 x C2
of Number Field in a with defining polynomial x^2 + 20072
sage: I = (G.0)^11; I
- Fractional ideal class (41, 1/2*a + 5)
+ Fractional ideal class (33, 1/2*a + 8)
sage: J = G(I.ideal()^5); J
- Fractional ideal class (115856201, 1/2*a + 40407883)
+ Fractional ideal class (39135393, 1/2*a + 13654253)
sage: J.reduce()
- Fractional ideal class (57, 1/2*a + 44)
+ Fractional ideal class (73, 1/2*a + 47)
sage: J == I^5
True
"""
diff --git a/src/sage/rings/number_field/galois_group.py b/src/sage/rings/number_field/galois_group.py
index 79acd053bb..e060148e4d 100644
--- a/src/sage/rings/number_field/galois_group.py
+++ b/src/sage/rings/number_field/galois_group.py
@@ -944,7 +944,7 @@ class GaloisGroup_v2(GaloisGroup_perm):
sage: K.<b> = NumberField(x^4 - 2*x^2 + 2, 'a').galois_closure()
sage: G = K.galois_group()
sage: [G.artin_symbol(P) for P in K.primes_above(7)]
- [(1,5)(2,6)(3,7)(4,8), (1,5)(2,6)(3,7)(4,8), (1,4)(2,3)(5,8)(6,7), (1,4)(2,3)(5,8)(6,7)]
+ [(1,4)(2,3)(5,8)(6,7), (1,4)(2,3)(5,8)(6,7), (1,5)(2,6)(3,7)(4,8), (1,5)(2,6)(3,7)(4,8)]
sage: G.artin_symbol(17)
Traceback (most recent call last):
...
diff --git a/src/sage/rings/number_field/number_field.py b/src/sage/rings/number_field/number_field.py
index 58463d570d..ff65634e99 100644
--- a/src/sage/rings/number_field/number_field.py
+++ b/src/sage/rings/number_field/number_field.py
@@ -3643,7 +3643,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: L.<b> = K.extension(x^2 - 3, x^2 + 1)
sage: M.<c> = L.extension(x^2 + 1)
sage: L.ideal(K.ideal(2, a))
- Fractional ideal (-a)
+ Fractional ideal (a)
sage: M.ideal(K.ideal(2, a)) == M.ideal(a*(b - c)/2)
True
@@ -4227,7 +4227,8 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
(y^2 + 6, Mod(1/6*y, y^2 + 6), Mod(6*y, y^2 + 1/6))
"""
f = self.absolute_polynomial()._pari_with_name('y')
- if f.pollead() == f.content().denominator() == 1:
+ f = f * f.content().denominator()
+ if f.pollead() == 1:
g = f
alpha = beta = g.variable().Mod(g)
else:
@@ -4821,7 +4822,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: K.<a> = NumberField(2*x^2 - 1/3)
sage: K._S_class_group_and_units(tuple(K.primes_above(2) + K.primes_above(3)))
- ([-6*a + 2, 6*a + 3, -1, 12*a + 5], [])
+ ([6*a + 2, 6*a + 3, -1, -12*a + 5], [])
"""
K_pari = self.pari_bnf(proof=proof)
S_pari = [p.pari_prime() for p in sorted(set(S))]
@@ -5166,7 +5167,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: [K.ideal(g).factor() for g in gens]
[(Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1)),
- Fractional ideal (-a),
+ Fractional ideal (a),
(Fractional ideal (2, a + 1))^2,
1]
@@ -5751,7 +5752,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: K.elements_of_norm(3)
[]
sage: K.elements_of_norm(50)
- [-7*a + 1, 5*a - 5, 7*a + 1]
+ [-a - 7, 5*a - 5, 7*a + 1]
TESTS:
@@ -5863,7 +5864,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: K.factor(1+a)
Fractional ideal (a + 1)
sage: K.factor(1+a/5)
- (Fractional ideal (a + 1)) * (Fractional ideal (-a - 2))^-1 * (Fractional ideal (2*a + 1))^-1 * (Fractional ideal (-3*a - 2))
+ (Fractional ideal (a + 1)) * (Fractional ideal (-a - 2))^-1 * (Fractional ideal (2*a + 1))^-1 * (Fractional ideal (-2*a + 3))
An example over a relative number field::
@@ -6460,9 +6461,9 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: new_basis = k.reduced_basis(prec=120)
sage: [c.minpoly() for c in new_basis]
[x - 1,
- x^2 - x + 1,
+ x^2 + x + 1,
+ x^6 + 3*x^5 - 102*x^4 - 103*x^3 + 10572*x^2 - 59919*x + 127657,
x^6 + 3*x^5 - 102*x^4 - 103*x^3 + 10572*x^2 - 59919*x + 127657,
- x^6 - 3*x^5 - 102*x^4 + 315*x^3 + 10254*x^2 - 80955*x + 198147,
x^3 - 171*x + 848,
x^6 + 171*x^4 + 1696*x^3 + 29241*x^2 + 145008*x + 719104]
sage: R = k.order(new_basis)
@@ -7058,7 +7059,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
-a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 + 2,
-2*a^16 + 3*a^15 - 3*a^14 + 3*a^13 - 3*a^12 + a^11 - a^9 + 3*a^8 - 4*a^7 + 5*a^6 - 6*a^5 + 4*a^4 - 3*a^3 + 2*a^2 + 2*a - 4,
a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,
- -a^14 - a^13 + a^12 + 2*a^10 + a^8 - 2*a^7 - 2*a^6 + 2*a^3 - a^2 + 2*a - 2)
+ 2*a^16 + a^15 - a^11 - 3*a^10 - 4*a^9 - 4*a^8 - 4*a^7 - 5*a^6 - 7*a^5 - 8*a^4 - 6*a^3 - 5*a^2 - 6*a - 7)
TESTS:
@@ -7067,7 +7068,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: K.<a> = NumberField(1/2*x^2 - 1/6)
sage: K.units()
- (-3*a + 2,)
+ (3*a - 2,)
"""
proof = proof_flag(proof)
@@ -7146,7 +7147,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
sage: U.gens()
(u0, u1, u2, u3, u4, u5, u6, u7, u8)
sage: U.gens_values() # result not independently verified
- [-1, -a^9 - a + 1, -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1, 2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4, a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1, -a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 + 2, -2*a^16 + 3*a^15 - 3*a^14 + 3*a^13 - 3*a^12 + a^11 - a^9 + 3*a^8 - 4*a^7 + 5*a^6 - 6*a^5 + 4*a^4 - 3*a^3 + 2*a^2 + 2*a - 4, a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2, -a^14 - a^13 + a^12 + 2*a^10 + a^8 - 2*a^7 - 2*a^6 + 2*a^3 - a^2 + 2*a - 2]
+ [-1, -a^9 - a + 1, -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1, 2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4, a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1, -a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 + 2, -2*a^16 + 3*a^15 - 3*a^14 + 3*a^13 - 3*a^12 + a^11 - a^9 + 3*a^8 - 4*a^7 + 5*a^6 - 6*a^5 + 4*a^4 - 3*a^3 + 2*a^2 + 2*a - 4, a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2, 2*a^16 + a^15 - a^11 - 3*a^10 - 4*a^9 - 4*a^8 - 4*a^7 - 5*a^6 - 7*a^5 - 8*a^4 - 6*a^3 - 5*a^2 - 6*a - 7]
"""
proof = proof_flag(proof)
diff --git a/src/sage/rings/number_field/number_field_element.pyx b/src/sage/rings/number_field/number_field_element.pyx
index 784c239dc1..aa740069dc 100644
--- a/src/sage/rings/number_field/number_field_element.pyx
+++ b/src/sage/rings/number_field/number_field_element.pyx
@@ -4446,7 +4446,7 @@ cdef class NumberFieldElement(FieldElement):
sage: f = Qi.embeddings(K)[0]
sage: a = f(2+3*i) * (2-zeta)^2
sage: a.descend_mod_power(Qi,2)
- [-3*i - 2, -2*i + 3]
+ [-2*i + 3, 3*i + 2]
An absolute example::
@@ -5124,7 +5124,7 @@ cdef class NumberFieldElement_relative(NumberFieldElement):
EXAMPLES::
sage: K.<a, b, c> = NumberField([x^2 - 2, x^2 - 3, x^2 - 5])
- sage: P = K.prime_factors(5)[0]
+ sage: P = K.prime_factors(5)[1]
sage: (2*a + b - c).valuation(P)
1
"""
diff --git a/src/sage/rings/number_field/number_field_ideal.py b/src/sage/rings/number_field/number_field_ideal.py
index 5f587556a4..33481fead0 100644
--- a/src/sage/rings/number_field/number_field_ideal.py
+++ b/src/sage/rings/number_field/number_field_ideal.py
@@ -3355,7 +3355,7 @@ def quotient_char_p(I, p):
[]
sage: I = K.factor(13)[0][0]; I
- Fractional ideal (-3*i - 2)
+ Fractional ideal (-2*i + 3)
sage: I.residue_class_degree()
1
sage: quotient_char_p(I, 13)[0]
diff --git a/src/sage/rings/number_field/number_field_ideal_rel.py b/src/sage/rings/number_field/number_field_ideal_rel.py
index bae36d4b9c..f64bd5b761 100644
--- a/src/sage/rings/number_field/number_field_ideal_rel.py
+++ b/src/sage/rings/number_field/number_field_ideal_rel.py
@@ -272,7 +272,7 @@ class NumberFieldFractionalIdeal_rel(NumberFieldFractionalIdeal):
sage: L.<b> = K.extension(5*x^2 + 1)
sage: P = L.primes_above(2)[0]
sage: P.gens_reduced()
- (2, 15*a*b + 3*a + 1)
+ (2, -15*a*b + 3*a + 1)
"""
try:
# Compute the single generator, if it exists
@@ -401,7 +401,7 @@ class NumberFieldFractionalIdeal_rel(NumberFieldFractionalIdeal):
sage: L.<b> = K.extension(5*x^2 + 1)
sage: P = L.primes_above(2)[0]
sage: P.relative_norm()
- Fractional ideal (-6*a + 2)
+ Fractional ideal (6*a + 2)
"""
L = self.number_field()
K = L.base_field()
@@ -518,7 +518,7 @@ class NumberFieldFractionalIdeal_rel(NumberFieldFractionalIdeal):
sage: L.<b> = K.extension(5*x^2 + 1)
sage: P = L.primes_above(2)[0]
sage: P.ideal_below()
- Fractional ideal (-6*a + 2)
+ Fractional ideal (6*a + 2)
"""
L = self.number_field()
K = L.base_field()
diff --git a/src/sage/rings/number_field/number_field_rel.py b/src/sage/rings/number_field/number_field_rel.py
index d33980c4b1..50e846b205 100644
--- a/src/sage/rings/number_field/number_field_rel.py
+++ b/src/sage/rings/number_field/number_field_rel.py
@@ -213,14 +213,14 @@ class NumberField_relative(NumberField_generic):
sage: l.<b> = k.extension(5*x^2 + 3); l
Number Field in b with defining polynomial 5*x^2 + 3 over its base field
sage: l.pari_rnf()
- [x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4), ..., y^4 + 6*y^2 + 1, x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4)], [0, 0]]
+ [x^2 + (-y^3 + 1/2*y^2 - 6*y + 3/2)*x + (-3/4*y^3 - 1/4*y^2 - 17/4*y - 19/4), ..., y^4 + 6*y^2 + 1, x^2 + (-y^3 + 1/2*y^2 - 6*y + 3/2)*x + (-3/4*y^3 - 1/4*y^2 - 17/4*y - 19/4)], [0, 0]]
sage: b
b
sage: l.<b> = k.extension(x^2 + 3/5); l
Number Field in b with defining polynomial x^2 + 3/5 over its base field
sage: l.pari_rnf()
- [x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4), ..., y^4 + 6*y^2 + 1, x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4)], [0, 0]]
+ [x^2 + (-y^3 + 1/2*y^2 - 6*y + 3/2)*x + (-3/4*y^3 - 1/4*y^2 - 17/4*y - 19/4), ..., y^4 + 6*y^2 + 1, x^2 + (-y^3 + 1/2*y^2 - 6*y + 3/2)*x + (-3/4*y^3 - 1/4*y^2 - 17/4*y - 19/4)], [0, 0]]
sage: b
b
diff --git a/src/sage/rings/number_field/order.py b/src/sage/rings/number_field/order.py
index 6eca89ed8d..78ef4c3b33 100644
--- a/src/sage/rings/number_field/order.py
+++ b/src/sage/rings/number_field/order.py
@@ -520,7 +520,7 @@ class Order(IntegralDomain, sage.rings.abc.Order):
sage: k.<a> = NumberField(x^2 + 5077); G = k.class_group(); G
Class group of order 22 with structure C22 of Number Field in a with defining polynomial x^2 + 5077
sage: G.0 ^ -9
- Fractional ideal class (11, a + 7)
+ Fractional ideal class (43, a + 13)
sage: Ok = k.maximal_order(); Ok
Maximal Order in Number Field in a with defining polynomial x^2 + 5077
sage: Ok * (11, a + 7)
diff --git a/src/sage/rings/number_field/selmer_group.py b/src/sage/rings/number_field/selmer_group.py
index c534aaa9f6..6bc67565d2 100644
--- a/src/sage/rings/number_field/selmer_group.py
+++ b/src/sage/rings/number_field/selmer_group.py
@@ -491,7 +491,7 @@ def pSelmerGroup(K, S, p, proof=None, debug=False):
sage: [K.ideal(g).factor() for g in gens]
[(Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1)),
- Fractional ideal (-a),
+ Fractional ideal (a),
(Fractional ideal (2, a + 1))^2,
1]
diff --git a/src/sage/rings/polynomial/polynomial_quotient_ring.py b/src/sage/rings/polynomial/polynomial_quotient_ring.py
index bb5d8356be..8a7e5fa66f 100644
--- a/src/sage/rings/polynomial/polynomial_quotient_ring.py
+++ b/src/sage/rings/polynomial/polynomial_quotient_ring.py
@@ -1791,7 +1791,7 @@ class PolynomialQuotientRing_generic(CommutativeRing):
sage: D.selmer_generators([K.ideal(2, -a+1), K.ideal(3, a+1)], 3)
[2, a + 1]
sage: D.selmer_generators([K.ideal(2, -a+1), K.ideal(3, a+1), K.ideal(a)], 3)
- [2, a + 1, a]
+ [2, a + 1, -a]
"""
fields, isos, iso_classes = self._S_decomposition(tuple(S))
diff --git a/src/sage/rings/qqbar.py b/src/sage/rings/qqbar.py
index 704b77ce5f..83ee4549e4 100644
--- a/src/sage/rings/qqbar.py
+++ b/src/sage/rings/qqbar.py
@@ -312,8 +312,8 @@ and we get a way to produce the number directly::
True
sage: sage_input(n)
R.<y> = QQ[]
- v = AA.polynomial_root(AA.common_polynomial(y^4 - 4*y^2 + 1), RIF(RR(0.51763809020504148), RR(0.51763809020504159)))
- -109*v^3 - 89*v^2 + 327*v + 178
+ v = AA.polynomial_root(AA.common_polynomial(y^4 - 4*y^2 + 1), RIF(-RR(1.9318516525781366), -RR(1.9318516525781364)))
+ -109*v^3 + 89*v^2 + 327*v - 178
We can also see that some computations (basically, those which are
easy to perform exactly) are performed directly, instead of storing
@@ -362,7 +362,7 @@ algorithms in :trac:`10255`::
# Verified
R1.<x> = QQbar[]
R2.<y> = QQ[]
- v = AA.polynomial_root(AA.common_polynomial(y^4 - 4*y^2 + 1), RIF(RR(0.51763809020504148), RR(0.51763809020504159)))
+ v = AA.polynomial_root(AA.common_polynomial(y^4 - 4*y^2 + 1), RIF(-RR(1.9318516525781366), -RR(1.9318516525781364)))
AA.polynomial_root(AA.common_polynomial(x^4 + QQbar(v^3 - 3*v - 1)*x^3 + QQbar(-v^3 + 3*v - 3)*x^2 + QQbar(-3*v^3 + 9*v + 3)*x + QQbar(3*v^3 - 9*v)), RIF(RR(0.99999999999999989), RR(1.0000000000000002)))
sage: one
1
@@ -2310,7 +2310,7 @@ def do_polred(poly, threshold=32):
cost = 2 * bitsize.nbits() + 5 * poly.degree().nbits()
if cost > threshold:
return parent.gen(), parent.gen(), poly
- new_poly, elt_back = poly.__pari__().polredbest(flag=1)
+ new_poly, elt_back = poly.numerator().__pari__().polredbest(flag=1)
elt_fwd = elt_back.modreverse()
return parent(elt_fwd.lift()), parent(elt_back.lift()), parent(new_poly)
@@ -2542,10 +2542,10 @@ def number_field_elements_from_algebraics(numbers, minimal=False, same_field=Fal
Defn: a |--> 1.414213562373095?)
sage: number_field_elements_from_algebraics((rt2,rt3))
- (Number Field in a with defining polynomial y^4 - 4*y^2 + 1, [-a^3 + 3*a, -a^2 + 2], Ring morphism:
+ (Number Field in a with defining polynomial y^4 - 4*y^2 + 1, [-a^3 + 3*a, a^2 - 2], Ring morphism:
From: Number Field in a with defining polynomial y^4 - 4*y^2 + 1
To: Algebraic Real Field
- Defn: a |--> 0.5176380902050415?)
+ Defn: a |--> -1.931851652578137?)
``rt3a`` is a real number in ``QQbar``. Ordinarily, we'd get a homomorphism
to ``AA`` (because all elements are real), but if we specify ``same_field=True``,
@@ -2570,7 +2570,7 @@ def number_field_elements_from_algebraics(numbers, minimal=False, same_field=Fal
(Number Field in a with defining polynomial y^4 - 4*y^2 + 1, -a^3 + 3*a, Ring morphism:
From: Number Field in a with defining polynomial y^4 - 4*y^2 + 1
To: Algebraic Real Field
- Defn: a |--> 0.5176380902050415?)
+ Defn: a |--> -1.931851652578137?)
We can specify ``minimal=True`` if we want the smallest number field::
@@ -2618,7 +2618,7 @@ def number_field_elements_from_algebraics(numbers, minimal=False, same_field=Fal
sage: nfI^2
-1
sage: sum = nfrt2 + nfrt3 + nfI + nfz3; sum
- 2*a^6 + a^5 - a^4 - a^3 - 2*a^2 - a
+ a^5 + a^4 - a^3 + 2*a^2 - a - 1
sage: hom(sum)
2.646264369941973? + 1.866025403784439?*I
sage: hom(sum) == rt2 + rt3 + qqI + z3
@@ -2658,7 +2658,7 @@ def number_field_elements_from_algebraics(numbers, minimal=False, same_field=Fal
sage: nf, nums, hom = number_field_elements_from_algebraics(elems, embedded=True)
sage: nf
Number Field in a with defining polynomial y^24 - 6*y^23 ...- 9*y^2 + 1
- with a = 0.2598678911433438? + 0.0572892247058457?*I
+ with a = 0.2598679? + 0.0572892?*I
sage: list(map(QQbar, nums)) == elems == list(map(hom, nums))
True
@@ -2725,7 +2725,7 @@ def number_field_elements_from_algebraics(numbers, minimal=False, same_field=Fal
sqrt(2), AA.polynomial_root(x^3-3, RIF(0,3)), 11/9, 1]
sage: res = number_field_elements_from_algebraics(my_nums, embedded=True)
sage: res[0]
- Number Field in a with defining polynomial y^24 - 107010*y^22 - 24*y^21 + ... + 250678447193040618624307096815048024318853254384 with a = -95.5053039433554?
+ Number Field in a with defining polynomial y^24 - 107010*y^22 - 24*y^21 + ... + 250678447193040618624307096815048024318853254384 with a = 93.32530798172420?
"""
gen = qq_generator
@@ -3129,7 +3129,7 @@ class AlgebraicGenerator(SageObject):
sage: root = ANRoot(x^2 - x - 1, RIF(1, 2))
sage: gen = AlgebraicGenerator(nf, root)
sage: gen.pari_field()
- [y^2 - y - 1, [2, 0], ...]
+ [[y^2 - y - 1, [2, 0], ...]
"""
if self.is_trivial():
raise ValueError("No PARI field attached to trivial generator")
@@ -3213,7 +3213,7 @@ class AlgebraicGenerator(SageObject):
sage: qq_generator.union(gen3) is gen3
True
sage: gen2.union(gen3)
- Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in 0.5176380902050415?
+ Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in -1.931851652578137?
"""
if self._trivial:
return other
@@ -3306,13 +3306,13 @@ class AlgebraicGenerator(SageObject):
Number Field in a with defining polynomial y^2 - 3 with a in 1.732050807568878?
sage: gen2_3 = gen2.union(gen3)
sage: gen2_3
- Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in 0.5176380902050415?
+ Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in -1.931851652578137?
sage: qq_generator.super_poly(gen2) is None
True
sage: gen2.super_poly(gen2_3)
-a^3 + 3*a
sage: gen3.super_poly(gen2_3)
- -a^2 + 2
+ a^2 - 2
"""
if checked is None:
@@ -3360,13 +3360,13 @@ class AlgebraicGenerator(SageObject):
sage: sqrt3 = ANExtensionElement(gen3, nf3.gen())
sage: gen2_3 = gen2.union(gen3)
sage: gen2_3
- Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in 0.5176380902050415?
+ Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in -1.931851652578137?
sage: gen2_3(sqrt2)
-a^3 + 3*a
sage: gen2_3(ANRational(1/7))
1/7
sage: gen2_3(sqrt3)
- -a^2 + 2
+ a^2 - 2
"""
if self._trivial:
return elt._value
@@ -4336,10 +4336,10 @@ class AlgebraicNumber_base(sage.structure.element.FieldElement):
sage: rt3 = AA(sqrt(3))
sage: rt3b = rt2 + rt3 - rt2
sage: rt3b.as_number_field_element()
- (Number Field in a with defining polynomial y^4 - 4*y^2 + 1, -a^2 + 2, Ring morphism:
+ (Number Field in a with defining polynomial y^4 - 4*y^2 + 1, a^2 - 2, Ring morphism:
From: Number Field in a with defining polynomial y^4 - 4*y^2 + 1
To: Algebraic Real Field
- Defn: a |--> 0.5176380902050415?)
+ Defn: a |--> -1.931851652578137?)
sage: rt3b.as_number_field_element(minimal=True)
(Number Field in a with defining polynomial y^2 - 3, a, Ring morphism:
From: Number Field in a with defining polynomial y^2 - 3
@@ -4401,7 +4401,7 @@ class AlgebraicNumber_base(sage.structure.element.FieldElement):
sage: rt2b = rt3 + rt2 - rt3
sage: rt2b.exactify()
sage: rt2b._exact_value()
- a^3 - 3*a where a^4 - 4*a^2 + 1 = 0 and a in 1.931851652578137?
+ a^3 - 3*a where a^4 - 4*a^2 + 1 = 0 and a in -0.5176380902050415?
sage: rt2b.simplify()
sage: rt2b._exact_value()
a where a^2 - 2 = 0 and a in 1.414213562373095?
@@ -4422,7 +4422,7 @@ class AlgebraicNumber_base(sage.structure.element.FieldElement):
sage: QQbar(2)._exact_field()
Trivial generator
sage: (sqrt(QQbar(2)) + sqrt(QQbar(19)))._exact_field()
- Number Field in a with defining polynomial y^4 - 20*y^2 + 81 with a in 2.375100220297941?
+ Number Field in a with defining polynomial y^4 - 20*y^2 + 81 with a in -3.789313782671036?
sage: (QQbar(7)^(3/5))._exact_field()
Number Field in a with defining polynomial y^5 - 2*y^4 - 18*y^3 + 38*y^2 + 82*y - 181 with a in 2.554256611698490?
"""
@@ -4442,7 +4442,7 @@ class AlgebraicNumber_base(sage.structure.element.FieldElement):
sage: QQbar(2)._exact_value()
2
sage: (sqrt(QQbar(2)) + sqrt(QQbar(19)))._exact_value()
- -1/9*a^3 - a^2 + 11/9*a + 10 where a^4 - 20*a^2 + 81 = 0 and a in 2.375100220297941?
+ -1/9*a^3 + a^2 + 11/9*a - 10 where a^4 - 20*a^2 + 81 = 0 and a in -3.789313782671036?
sage: (QQbar(7)^(3/5))._exact_value()
2*a^4 + 2*a^3 - 34*a^2 - 17*a + 150 where a^5 - 2*a^4 - 18*a^3 + 38*a^2 + 82*a - 181 = 0 and a in 2.554256611698490?
"""
@@ -6857,7 +6857,7 @@ class AlgebraicPolynomialTracker(SageObject):
sage: p = sqrt(AA(2)) * x^2 - sqrt(AA(3))
sage: cp = AA.common_polynomial(p)
sage: cp.generator()
- Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in 1.931851652578137?
+ Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in -0.5176380902050415?
"""
self.exactify()
return self._gen
@@ -7706,7 +7706,7 @@ class ANExtensionElement(ANDescr):
sage: rt2b.exactify()
sage: rt2b._descr
- a^3 - 3*a where a^4 - 4*a^2 + 1 = 0 and a in 1.931851652578137?
+ a^3 - 3*a where a^4 - 4*a^2 + 1 = 0 and a in -0.5176380902050415?
sage: rt2b._descr.is_simple()
False
"""
@@ -7791,7 +7791,7 @@ class ANExtensionElement(ANDescr):
sage: rt2b = rt3 + rt2 - rt3
sage: rt2b.exactify()
sage: rt2b._descr
- a^3 - 3*a where a^4 - 4*a^2 + 1 = 0 and a in 1.931851652578137?
+ a^3 - 3*a where a^4 - 4*a^2 + 1 = 0 and a in -0.5176380902050415?
sage: rt2b._descr.simplify(rt2b)
a where a^2 - 2 = 0 and a in 1.414213562373095?
"""
@@ -7830,9 +7830,9 @@ class ANExtensionElement(ANDescr):
sage: type(b)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: b.neg(a)
- 1/3*a^3 - 2/3*a^2 + 4/3*a - 2 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in -0.7247448713915890? - 1.573132184970987?*I
+ -1/3*a^3 + 1/3*a^2 - a - 1 where a^4 - 2*a^3 + a^2 + 6*a + 3 = 0 and a in 1.724744871391589? + 1.573132184970987?*I
sage: b.neg("ham spam and eggs")
- 1/3*a^3 - 2/3*a^2 + 4/3*a - 2 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in -0.7247448713915890? - 1.573132184970987?*I
+ -1/3*a^3 + 1/3*a^2 - a - 1 where a^4 - 2*a^3 + a^2 + 6*a + 3 = 0 and a in 1.724744871391589? + 1.573132184970987?*I
"""
return ANExtensionElement(self._generator, -self._value)
@@ -7848,9 +7848,9 @@ class ANExtensionElement(ANDescr):
sage: type(b)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: b.invert(a)
- 7/3*a^3 - 2/3*a^2 + 4/3*a - 12 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in -0.7247448713915890? - 1.573132184970987?*I
+ -7/3*a^3 + 19/3*a^2 - 7*a - 9 where a^4 - 2*a^3 + a^2 + 6*a + 3 = 0 and a in 1.724744871391589? + 1.573132184970987?*I
sage: b.invert("ham spam and eggs")
- 7/3*a^3 - 2/3*a^2 + 4/3*a - 12 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in -0.7247448713915890? - 1.573132184970987?*I
+ -7/3*a^3 + 19/3*a^2 - 7*a - 9 where a^4 - 2*a^3 + a^2 + 6*a + 3 = 0 and a in 1.724744871391589? + 1.573132184970987?*I
"""
return ANExtensionElement(self._generator, ~self._value)
@@ -7866,9 +7866,9 @@ class ANExtensionElement(ANDescr):
sage: type(b)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: b.conjugate(a)
- -1/3*a^3 + 2/3*a^2 - 4/3*a + 2 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in -0.7247448713915890? + 1.573132184970987?*I
+ 1/3*a^3 - 1/3*a^2 + a + 1 where a^4 - 2*a^3 + a^2 + 6*a + 3 = 0 and a in 1.724744871391589? - 1.573132184970987?*I
sage: b.conjugate("ham spam and eggs")
- -1/3*a^3 + 2/3*a^2 - 4/3*a + 2 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in -0.7247448713915890? + 1.573132184970987?*I
+ 1/3*a^3 - 1/3*a^2 + a + 1 where a^4 - 2*a^3 + a^2 + 6*a + 3 = 0 and a in 1.724744871391589? - 1.573132184970987?*I
"""
if self._exactly_real:
return self
@@ -8501,7 +8501,7 @@ def an_binop_expr(a, b, op):
sage: x = an_binop_expr(a, b, operator.add); x
<sage.rings.qqbar.ANBinaryExpr object at ...>
sage: x.exactify()
- -6/7*a^7 + 2/7*a^6 + 71/7*a^5 - 26/7*a^4 - 125/7*a^3 + 72/7*a^2 + 43/7*a - 47/7 where a^8 - 12*a^6 + 23*a^4 - 12*a^2 + 1 = 0 and a in 3.12580...?
+ 6/7*a^7 - 2/7*a^6 - 71/7*a^5 + 26/7*a^4 + 125/7*a^3 - 72/7*a^2 - 43/7*a + 47/7 where a^8 - 12*a^6 + 23*a^4 - 12*a^2 + 1 = 0 and a in -0.3199179336182997?
sage: a = QQbar(sqrt(2)) + QQbar(sqrt(3))
sage: b = QQbar(sqrt(3)) + QQbar(sqrt(5))
@@ -8510,7 +8510,7 @@ def an_binop_expr(a, b, op):
sage: x = an_binop_expr(a, b, operator.mul); x
<sage.rings.qqbar.ANBinaryExpr object at ...>
sage: x.exactify()
- 2*a^7 - a^6 - 24*a^5 + 12*a^4 + 46*a^3 - 22*a^2 - 22*a + 9 where a^8 - 12*a^6 + 23*a^4 - 12*a^2 + 1 = 0 and a in 3.1258...?
+ 2*a^7 - a^6 - 24*a^5 + 12*a^4 + 46*a^3 - 22*a^2 - 22*a + 9 where a^8 - 12*a^6 + 23*a^4 - 12*a^2 + 1 = 0 and a in -0.3199179336182997?
"""
return ANBinaryExpr(a, b, op)
diff --git a/src/sage/schemes/affine/affine_morphism.py b/src/sage/schemes/affine/affine_morphism.py
index 1c4f2dff18..32c2e47e49 100644
--- a/src/sage/schemes/affine/affine_morphism.py
+++ b/src/sage/schemes/affine/affine_morphism.py
@@ -1148,9 +1148,9 @@ class SchemeMorphism_polynomial_affine_space_field(SchemeMorphism_polynomial_aff
sage: H = End(A)
sage: f = H([(QQbar(sqrt(2))*x^2 + 1/QQbar(sqrt(3))) / (5*x)])
sage: f.reduce_base_field()
- Scheme endomorphism of Affine Space of dimension 1 over Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a = 1.931851652578137?
+ Scheme endomorphism of Affine Space of dimension 1 over Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a = ...?
Defn: Defined on coordinates by sending (x) to
- (((a^3 - 3*a)*x^2 + (1/3*a^2 - 2/3))/(5*x))
+ (((a^3 - 3*a)*x^2 + (-1/3*a^2 + 2/3))/(5*x))
::
diff --git a/src/sage/schemes/elliptic_curves/ell_field.py b/src/sage/schemes/elliptic_curves/ell_field.py
index 68b8375dae..48f358ea6a 100644
--- a/src/sage/schemes/elliptic_curves/ell_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_field.py
@@ -845,7 +845,7 @@ class EllipticCurve_field(ell_generic.EllipticCurve_generic, ProjectivePlaneCurv
sage: E = E.base_extend(G).quadratic_twist(c); E
Elliptic Curve defined by y^2 = x^3 + 5*a0*x^2 + (-200*a0^2)*x + (-42000*a0^2+42000*a0+126000) over Number Field in a0 with defining polynomial x^3 - 3*x^2 + 3*x + 9
sage: K.<b> = E.division_field(3, simplify_all=True); K
- Number Field in b with defining polynomial x^12 - 10*x^10 + 55*x^8 - 60*x^6 + 75*x^4 + 1350*x^2 + 2025
+ Number Field in b with defining polynomial x^12 + 5*x^10 + 40*x^8 + 315*x^6 + 750*x^4 + 675*x^2 + 2025
Some higher-degree examples::
diff --git a/src/sage/schemes/elliptic_curves/ell_generic.py b/src/sage/schemes/elliptic_curves/ell_generic.py
index 926ae310ea..3bae819fb0 100644
--- a/src/sage/schemes/elliptic_curves/ell_generic.py
+++ b/src/sage/schemes/elliptic_curves/ell_generic.py
@@ -3324,8 +3324,8 @@ class EllipticCurve_generic(WithEqualityById, plane_curve.ProjectivePlaneCurve):
sage: K.<a> = QuadraticField(2)
sage: E = EllipticCurve([1,a])
sage: E.pari_curve()
- [Mod(0, y^2 - 2), Mod(0, y^2 - 2), Mod(0, y^2 - 2), Mod(1, y^2 - 2),
- Mod(y, y^2 - 2), Mod(0, y^2 - 2), Mod(2, y^2 - 2), Mod(4*y, y^2 - 2),
+ [0, 0, 0, Mod(1, y^2 - 2),
+ Mod(y, y^2 - 2), 0, Mod(2, y^2 - 2), Mod(4*y, y^2 - 2),
Mod(-1, y^2 - 2), Mod(-48, y^2 - 2), Mod(-864*y, y^2 - 2),
Mod(-928, y^2 - 2), Mod(3456/29, y^2 - 2), Vecsmall([5]),
[[y^2 - 2, [2, 0], 8, 1, [[1, -1.41421356237310;
diff --git a/src/sage/schemes/elliptic_curves/ell_number_field.py b/src/sage/schemes/elliptic_curves/ell_number_field.py
index edbd196090..c44c803aa8 100644
--- a/src/sage/schemes/elliptic_curves/ell_number_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_number_field.py
@@ -218,7 +218,7 @@ class EllipticCurve_number_field(EllipticCurve_field):
sage: E == loads(dumps(E))
True
sage: E.simon_two_descent()
- (2, 2, [(0 : 0 : 1)])
+ (2, 2, [(0 : 0 : 1), (1/18*a + 7/18 : -5/54*a - 17/54 : 1)])
sage: E.simon_two_descent(lim1=5, lim3=5, limtriv=10, maxprob=7, limbigprime=10)
(2, 2, [(-1 : 0 : 1), (-2 : -1/2*a - 1/2 : 1)])
@@ -274,7 +274,7 @@ class EllipticCurve_number_field(EllipticCurve_field):
sage: E.simon_two_descent() # long time (4s on sage.math, 2013)
(3,
3,
- [(5/8*zeta43_0^2 + 17/8*zeta43_0 - 9/4 : -27/16*zeta43_0^2 - 103/16*zeta43_0 + 39/8 : 1),
+ [(1/8*zeta43_0^2 - 3/8*zeta43_0 - 1/4 : -5/16*zeta43_0^2 + 7/16*zeta43_0 + 1/8 : 1),
(0 : 0 : 1)])
"""
verbose = int(verbose)
@@ -865,7 +865,7 @@ class EllipticCurve_number_field(EllipticCurve_field):
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1,
- Local data at Fractional ideal (-3*i - 2):
+ Local data at Fractional ideal (-2*i + 3):
Reduction type: bad split multiplicative
Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3 over Number Field in i with defining polynomial x^2 + 1
Minimal discriminant valuation: 2
@@ -2645,12 +2645,12 @@ class EllipticCurve_number_field(EllipticCurve_field):
[-92, -23, -23]
sage: C.matrix() # long time
- [1 2 2 4 2 4]
- [2 1 2 2 4 4]
- [2 2 1 4 4 2]
- [4 2 4 1 3 3]
- [2 4 4 3 1 3]
- [4 4 2 3 3 1]
+ [1 2 2 4 4 2]
+ [2 1 2 4 2 4]
+ [2 2 1 2 4 4]
+ [4 4 2 1 3 3]
+ [4 2 4 3 1 3]
+ [2 4 4 3 3 1]
The graph of this isogeny class has a shape which does not
occur over `\QQ`: a triangular prism. Note that for curves
@@ -2677,12 +2677,12 @@ class EllipticCurve_number_field(EllipticCurve_field):
sage: G = C.graph() # long time
sage: G.adjacency_matrix() # long time
- [0 1 1 0 1 0]
- [1 0 1 1 0 0]
- [1 1 0 0 0 1]
- [0 1 0 0 1 1]
- [1 0 0 1 0 1]
- [0 0 1 1 1 0]
+ [0 1 1 0 0 1]
+ [1 0 1 0 1 0]
+ [1 1 0 1 0 0]
+ [0 0 1 0 1 1]
+ [0 1 0 1 0 1]
+ [1 0 0 1 1 0]
To display the graph without any edge labels::
@@ -3316,7 +3316,7 @@ class EllipticCurve_number_field(EllipticCurve_field):
sage: points = [E.lift_x(x) for x in xi]
sage: newpoints, U = E.lll_reduce(points) # long time (35s on sage.math, 2011)
sage: [P[0] for P in newpoints] # long time
- [6823803569166584943, 5949539878899294213, 2005024558054813068, 5864879778877955778, 23955263915878682727/4, 5922188321411938518, 5286988283823825378, 175620639884534615751/25, -11451575907286171572, 3502708072571012181, 1500143935183238709184/225, 27180522378120223419/4, -5811874164190604461581/625, 26807786527159569093, 7404442636649562303, 475656155255883588, 265757454726766017891/49, 7272142121019825303, 50628679173833693415/4, 6951643522366348968, 6842515151518070703, 111593750389650846885/16, 2607467890531740394315/9, -1829928525835506297]
+ [6823803569166584943, 5949539878899294213, 2005024558054813068, 5864879778877955778, 23955263915878682727/4, 5922188321411938518, 5286988283823825378, 11465667352242779838, -11451575907286171572, 3502708072571012181, 1500143935183238709184/225, 27180522378120223419/4, -5811874164190604461581/625, 26807786527159569093, 7041412654828066743, 475656155255883588, 265757454726766017891/49, 7272142121019825303, 50628679173833693415/4, 6951643522366348968, 6842515151518070703, 111593750389650846885/16, 2607467890531740394315/9, -1829928525835506297]
An example to show the explicit use of the height pairing matrix::
diff --git a/src/sage/schemes/elliptic_curves/ell_rational_field.py b/src/sage/schemes/elliptic_curves/ell_rational_field.py
index 3808822812..a75290ea35 100644
--- a/src/sage/schemes/elliptic_curves/ell_rational_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_rational_field.py
@@ -1827,7 +1827,7 @@ class EllipticCurve_rational_field(EllipticCurve_number_field):
sage: E = EllipticCurve('389a1')
sage: E._known_points = [] # clear cached points
sage: E.simon_two_descent()
- (2, 2, [(1 : 0 : 1), (-11/9 : 28/27 : 1)])
+ (2, 2, [(5/4 : 5/8 : 1), (-3/4 : 7/8 : 1)])
sage: E = EllipticCurve('5077a1')
sage: E.simon_two_descent()
(3, 3, [(1 : 0 : 1), (2 : 0 : 1), (0 : 2 : 1)])
diff --git a/src/sage/schemes/elliptic_curves/gal_reps_number_field.py b/src/sage/schemes/elliptic_curves/gal_reps_number_field.py
index 81ad295160..d484a4a18b 100644
--- a/src/sage/schemes/elliptic_curves/gal_reps_number_field.py
+++ b/src/sage/schemes/elliptic_curves/gal_reps_number_field.py
@@ -780,12 +780,12 @@ def deg_one_primes_iter(K, principal_only=False):
[Fractional ideal (2, a + 1),
Fractional ideal (3, a + 1),
Fractional ideal (3, a + 2),
- Fractional ideal (-a),
+ Fractional ideal (a),
Fractional ideal (7, a + 3),
Fractional ideal (7, a + 4)]
sage: it = deg_one_primes_iter(K, True)
sage: [next(it) for _ in range(6)]
- [Fractional ideal (-a),
+ [Fractional ideal (a),
Fractional ideal (-2*a + 3),
Fractional ideal (2*a + 3),
Fractional ideal (a + 6),
diff --git a/src/sage/schemes/elliptic_curves/gp_simon.py b/src/sage/schemes/elliptic_curves/gp_simon.py
index 28b97f34af..9f7d1b6020 100644
--- a/src/sage/schemes/elliptic_curves/gp_simon.py
+++ b/src/sage/schemes/elliptic_curves/gp_simon.py
@@ -56,7 +56,7 @@ def simon_two_descent(E, verbose=0, lim1=None, lim3=None, limtriv=None,
sage: import sage.schemes.elliptic_curves.gp_simon
sage: E=EllipticCurve('389a1')
sage: sage.schemes.elliptic_curves.gp_simon.simon_two_descent(E)
- (2, 2, [(1 : 0 : 1), (-11/9 : 28/27 : 1)])
+ (2, 2, [(5/4 : 5/8 : 1), (-3/4 : 7/8 : 1)])
TESTS::
@@ -117,7 +117,7 @@ def simon_two_descent(E, verbose=0, lim1=None, lim3=None, limtriv=None,
# The block below mimics the defaults in Simon's scripts, and needs to be changed
# when these are updated.
if K is QQ:
- cmd = 'ellrank(%s, %s);' % (list(E.ainvs()), [P.__pari__() for P in known_points])
+ cmd = 'ellQ_ellrank(%s, %s);' % (list(E.ainvs()), [P.__pari__() for P in known_points])
if lim1 is None:
lim1 = 5
if lim3 is None:
@@ -144,7 +144,7 @@ def simon_two_descent(E, verbose=0, lim1=None, lim3=None, limtriv=None,
if verbose > 0:
print(s)
v = gp.eval('ans')
- if v=='ans': # then the call to ellrank() or bnfellrank() failed
+ if v=='ans': # then the call to ellQ_ellrank() or bnfellrank() failed
raise RuntimeError("An error occurred while running Simon's 2-descent program")
if verbose >= 2:
print("v = %s" % v)
diff --git a/src/sage/schemes/elliptic_curves/isogeny_small_degree.py b/src/sage/schemes/elliptic_curves/isogeny_small_degree.py
index a936deb74f..dc19254d8c 100644
--- a/src/sage/schemes/elliptic_curves/isogeny_small_degree.py
+++ b/src/sage/schemes/elliptic_curves/isogeny_small_degree.py
@@ -1208,14 +1208,14 @@ def isogenies_13_0(E, minimal_models=True):
sage: [phi.codomain().ainvs() for phi in isogenies_13_0(E)] # long time (4s)
[(0,
0,
- 20360599/165164973653422080*a^11 - 3643073/41291243413355520*a^10 - 101/8789110986240*a^9 + 5557619461/573489491852160*a^8 - 82824971/11947697746920*a^7 - 19487/21127670640*a^6 - 475752603733/29409717530880*a^5 + 87205112531/7352429382720*a^4 + 8349/521670880*a^3 + 5858744881/12764634345*a^2 - 1858703809/2836585410*a + 58759402/48906645,
+ 20360599/165164973653422080*a^11 - 3643073/41291243413355520*a^10 + 1887439/1146978983704320*a^9 + 5557619461/573489491852160*a^8 - 82824971/11947697746920*a^7 + 1030632647/7965131831280*a^6 - 475752603733/29409717530880*a^5 + 87205112531/7352429382720*a^4 - 43618899433/204234149520*a^3 + 5858744881/12764634345*a^2 - 1858703809/2836585410*a + 2535050171/1418292705,
-139861295/2650795873449984*a^11 - 3455957/5664093746688*a^10 - 345310571/50976843720192*a^9 - 500530795/118001953056*a^8 - 12860048113/265504394376*a^7 - 25007420461/44250732396*a^6 + 458134176455/1416023436672*a^5 + 16701880631/9077073312*a^4 + 155941666417/9077073312*a^3 + 3499310115/378211388*a^2 - 736774863/94552847*a - 21954102381/94552847,
- 579363345221/13763747804451840*a^11 + 371192377511/860234237778240*a^10 + 8855090365657/1146978983704320*a^9 + 5367261541663/1633873196160*a^8 + 614883554332193/15930263662560*a^7 + 30485197378483/68078049840*a^6 - 131000897588387/2450809794240*a^5 - 203628705777949/306351224280*a^4 - 1587619388190379/204234149520*a^3 + 14435069706551/11346341640*a^2 + 7537273048614/472764235*a + 89198980034806/472764235),
+ 8342795944891/198197968384106496*a^11 + 8908625263589/20645621706677760*a^10 + 53130542636623/6881873902225920*a^9 + 376780111042213/114697898370432*a^8 + 614884052146333/15930263662560*a^7 + 3566768133324359/7965131831280*a^6 - 1885593809102545/35291661037056*a^5 - 2443732172026523/3676214691360*a^4 - 9525729503937541/1225404897120*a^3 + 51990274442321/40846829904*a^2 + 67834019370596/4254878115*a + 267603083706812/1418292705),
(0,
0,
- 20360599/165164973653422080*a^11 - 3643073/41291243413355520*a^10 - 101/8789110986240*a^9 + 5557619461/573489491852160*a^8 - 82824971/11947697746920*a^7 - 19487/21127670640*a^6 - 475752603733/29409717530880*a^5 + 87205112531/7352429382720*a^4 + 8349/521670880*a^3 + 5858744881/12764634345*a^2 - 1858703809/2836585410*a + 58759402/48906645,
+ 20360599/165164973653422080*a^11 - 3643073/41291243413355520*a^10 + 1887439/1146978983704320*a^9 + 5557619461/573489491852160*a^8 - 82824971/11947697746920*a^7 + 1030632647/7965131831280*a^6 - 475752603733/29409717530880*a^5 + 87205112531/7352429382720*a^4 - 43618899433/204234149520*a^3 + 5858744881/12764634345*a^2 - 1858703809/2836585410*a + 2535050171/1418292705,
-6465569317/1325397936724992*a^11 - 112132307/1960647835392*a^10 - 17075412917/25488421860096*a^9 - 207832519229/531008788752*a^8 - 1218275067617/265504394376*a^7 - 9513766502551/177002929584*a^6 + 4297077855437/708011718336*a^5 + 354485975837/4538536656*a^4 + 4199379308059/4538536656*a^3 - 30841577919/189105694*a^2 - 181916484042/94552847*a - 2135779171614/94552847,
- -132601797212627/3440936951112960*a^11 - 6212467020502021/13763747804451840*a^10 - 1515926454902497/286744745926080*a^9 - 15154913741799637/4901619588480*a^8 - 576888119803859263/15930263662560*a^7 - 86626751639648671/204234149520*a^6 + 16436657569218427/306351224280*a^5 + 1540027900265659087/2450809794240*a^4 + 375782662805915809/51058537380*a^3 - 14831920924677883/11346341640*a^2 - 7237947774817724/472764235*a - 84773764066089509/472764235)]
+ -1316873026840277/34172063514501120*a^11 - 18637401045099413/41291243413355520*a^10 - 36382234917217247/6881873902225920*a^9 - 61142238484016213/19775499719040*a^8 - 576888119306045123/15930263662560*a^7 - 3378443313906256321/7965131831280*a^6 + 326466167429333279/6084769144320*a^5 + 4620083325391594991/7352429382720*a^4 + 9018783894167184149/1225404897120*a^3 - 9206015742300283/7042556880*a^2 - 65141531411426446/4254878115*a - 254321286054666133/1418292705)]
"""
if E.j_invariant()!=0:
raise ValueError("j-invariant must be 0.")
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