sd18: gal image tutorial

902 days ago by pub

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')
(False, '5-torsion')
E.is_surjective(7) 
        
(True, None)
(True, None)
E.is_surjective(7, A=5) 
        
(False, [-1])
(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
True
E.is_reducible(7) 
        
False
False
E.is_irreducible(5) 
        
False
False

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