1205-Tutorial: Images of Galois in Sage.sagews

Tutorial: Images of Galois in Sage

E = EllipticCurve('11a')

Any prime $p$ not in the output of E.non_surjective() is such that $E[p]$ is provably surjective.

E.non_surjective()

One can ask about surjectivity for a specific prime.  If the output is Yes, then the representation is definitely surjective.  If no, then in some rare case (depending on the optional parameter $A$), it could still be surjective.

E.is_surjective(5)

(False, '5-torsion')

E.is_surjective(7)

(True, None)

E.is_surjective(7, A=5)

(False, [-1])

We can also check for reducibility, which is currently (lazily!) determined by simply computing the isogeny class and looking at the degrees that appear (dumb!).

E.is_reducible(5)

True

E.is_reducible(7)

False

E.is_irreducible(5)

False

A table of images:

for E in cremona_optimal_curves([1..100]):
    print E.cremona_label(), E.non_surjective()