### Realisation of Galois groups over Q via mod p modular forms

Every mod p eigenform gives rise to a two-dimensional odd Galois
representation in characteristic p. The ramified primes different from p
of this representation are given by the primes dividing the level.

If p is odd and the weight is one, then the representation is
known to be unramified at p. For p = 2, the same is strongly suspected,
but not known in some exceptional case.
If on the other hand the weight is bigger than one, one has to
expect ramification at p.

In many cases one can actually compute the image of the Galois
representation.

Using the implementation *Weight1*
of the computation of Hecke algebras in weight one over finite fields,
it is possible to realize many of the groups SL(2,2^r) as Galois
groups over Q. One can in general not give a polynomial, but knows,
which primes can ramify.

Here is the table of groups obtained. It can be either downloaded
as a pdf file or as a
Magma table.
The usage of the latter is the following. First load the table:

` load "GroupTable.mg"; `

If you want to access the data of field degree 2^r, type:

`GroupTable[r];`

Last modification: 9 January 2004.