On this page, you can download data and tables
describing all of the Galois representations modulo ℓ attached to
eigenforms of level 1 and weight k<ℓ which have residual degree 1 and
whose image contains SL_{2}(**F**_{ℓ}) for ℓ up to
31, as well as a few others, including three representations attached to
forms which admit a companion form mod ℓ. These forms are specified by
their LMFDB label.

These data were computed by the algorithm described in my article Computing modular Galois representations, and (when ℓ≤31) certified by the method described in my article Certification of Galois representations. The three representations attached to companion forms were studied in my article Companion forms and explicit computation of PGL_{2} number fields with very little ramification.

For each Galois representation, the data available for download below here are formed of

- an irreducible polynomial f(x) in
**Z**[x], - an ordered list of the roots of f(x) in the field
**F**_{p}[x]/f(x), where p is a prime such that f(x) is irreducible mod p, - and of a list of resolvents which can be used to compute the image
of Frobenius elements by the representation, along with the polynomial
h(x)∈
**Z**[x] used to compute them (cf. my articles for details).

The splitting field of f(x) is the smallest number field through
which the representation modulo S factors, where S is the largest
subgroup of **F**_{ℓ}^{*} which does not contain -1. The roots of f(x) correspond via the Galois representation to the points of (**F**_{ℓ}^{2} -{0})/S, and are ordered as follows:

(0,1), (0,2), ..., (0,ℓ-1), (1,0), ε(0,1), ε(0,2), ..., ε(0,ℓ-1), ε(1,0), ε^{2}(0,1), ε^{2}(0,2), ..., ε^{2}(0,ℓ-1), ε^{2}(1,0),...

where ε is the smallest positive integer which generates the multiplicative group of **F**_{ℓ}.

The resolvents each come with a tag which indicates the conjugacy class of GL_{2}(**F**_{ℓ})/S it corresponds to:

Tag | Conjugacy class |
---|---|

"Scal",a | Scalar matrix with double eigenvalue a |

"Diag",[a,b] | Split semisimple matrices with eigenvalues a and b |

"Irr",[t,d] | Nonsplit semisimple matrices with trace t and determinant d |

"Nil",a | Nonsemisimple matrices with double eigenvalue a. |

The data are provided in the following format:

[f(x), [root_{1},root_{2},...], h(x), [[resolvent_{1},tag_{1}],[resolvent_{2},tag_{2}],...]].

They have been rigorously certified unless explicit mention of the contrary.

- ℓ=11
- ℓ=13
- ℓ=17
- ℓ=19
- ℓ=23
- ℓ=29
- ℓ=31
- ℓ=37
- E
_{8}Δ (uncertified)

- E
- ℓ=41
- ℓ=43

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