Keywords: 389

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# Discriminant of the Hecke algebra

The discriminant of the Hecke algebra is the discriminant with respect to the trace pairing. The ring is a product of totally real number fields. Let denote the integral closure of T in ; note that where is the ring of integers of Ki. The discriminant of T also equals .

We view the primes dividing as arising from singularities of the individual irreducible components of . The primes dividing are viewed as arising from intersections of irreducible components, or what is the same, congruences between eigenforms.

The discriminant of the Hecke algebra T associated to is

This discriminant was computed by applying the definition of discriminant to a representation of the first 65 Hecke operators . Matrices representing these Hecke operators were computed using the modular symbols algorithms described in [Cremona, Algorithms for modular elliptic curves]. The number 65=390/6 of Hecke operators needed to generate Tas a Z-module was chosen in accord with the bound in [Sturm, On the congruence of modular forms].

In the case of X0(389), where Kd has degree d over Q. We have

The discriminants of the Ki are

K1 | K2 | K3 | K6 | K20
1 | 23 |  |  |

Observe that the discriminant of K20 is divisible by 389; this is the unusual behavior alluded to in the introduction. The product of the discriminants is

This differs from the exact discriminant by a factor of , so the index of T in its normalization is

Notice that 389 does not divide this index, and that 389 is not a ``congruence prime''; thus 389 does not divide any modular degrees. We do not know of any examples of optimal quotients of J0(p) such that p divides the modular degree.

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