Images of Galois

William Stein

April 2005

This is a table that gives information about the images of Galois representations attached to elliptic curves. 
     surj.1-30000       (10MB)
 
     surj.1-30000.bz2   (1MB)
For each optimal curve of conductor <= 30000, it gives a list of primes where the mod-p Galois representation is (or in a few cases, probably is) not surjective along with some information about why. Whenever a prime is not listed, that means the representation definitely is surjective. The notation is 
   conductor letter  number(=1)  rank  [a1,a2,a3,a4,a6]    [(p,why), ...]
The possibilities for why: 
  cm -- the curve has CM  (we do nothing further here at all, and 
        list (0,'cm') to mean "all primes")

  p-torsion -- mod-p repn. not surjective because there is p-torsion

  reducible_3-divpol -- the 3-division polynomial is reducible

  3-divpoly_galgroup_order_n -- the Galois group of 3-div 
                        poly has order n < 24

  [1] -- for all a_ell with certain properties, the poly 
            x^2 - a_ell x + ell (mod p) factors.  (ell<1000)

  [-1] -- for all a_ell with certain properties, the poly 
            x^2 - a_ell x + ell (mod p) is irreducible.  (ell<1000)
          There are only 4-cases like this; they are at level 675.
I created this table using the following program, which makes use of SAGE: 
       surj.tar

The main theoretical inputs to this computation are the following:

  1. Explicit 2 and 3-division polynomials.
  2. Results of Serre on images of Galois representations from his Inventiones paper.
  3. An effective bound B such that for p>=B the representation rho_{E,p} is surjective. This is from a paper of Cojocaru.
I intend to write up more about the theory and results for a paper. When I do so, I will add a link here.