Denis Simon’s PARI scripts¶
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sage.schemes.elliptic_curves.gp_simon.
init
()¶ Function to initialize the gp process
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sage.schemes.elliptic_curves.gp_simon.
simon_two_descent
(E, verbose=0, lim1=None, lim3=None, limtriv=None, maxprob=20, limbigprime=30, known_points=[])¶ Interface to Simon’s gp script for two-descent.
Note
Users should instead run E.simon_two_descent()
EXAMPLES:
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)])
TESTS:
sage: E = EllipticCurve('37a1').change_ring(QuadraticField(-11,'x')) sage: E.simon_two_descent() (1, 1, [(0 : 0 : 1)])
An example with an elliptic curve defined over a relative number field:
sage: F.<a> = QuadraticField(29) sage: x = QQ['x'].gen() sage: K.<b> = F.extension(x^2-1/2*a+1/2) sage: E = EllipticCurve(K,[1, 0, 5/2*a + 27/2, 0, 0]) # long time (about 3 s) sage: E.simon_two_descent(lim1=2, limtriv=3) (1, 1, ...)
Check that trac ticket #16022 is fixed:
sage: K.<y> = NumberField(x^4 + x^2 - 7) sage: E = EllipticCurve(K, [1, 0, 5*y^2 + 16, 0, 0]) sage: E.simon_two_descent(lim1=2, limtriv=3) # long time (about 3 s) (1, 1, ...)
An example that checks that trac ticket #9322 is fixed (it should take less than a second to run):
sage: K.<w> = NumberField(x^2-x-232) sage: E = EllipticCurve([2-w,18+3*w,209+9*w,2581+175*w,852-55*w]) sage: E.simon_two_descent() (0, 2, [])