#
Images of Galois

### 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:

- Explicit 2 and 3-division polynomials.
- Results of Serre on images of Galois representations from his Inventiones paper.
- 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.