Discriminant of the Hecke algebra

The discriminant of the Hecke algebra $\mathbf{T}\subset\mbox{\rm End}(S_2(\Gamma_0(N)))$ is the discriminant with respect to the trace pairing. The ring $\mathbf{T}\otimes\mathbf{Q}$ is a product $K_1\times\cdots\times K_n$ of totally real number fields. Let $\tilde{\mathbf{T}}$ denote the integral closure of  T in $\mathbf{T}\otimes\mathbf{Q}$; note that $\tilde{\mathbf{T}}=\prod \O_i$ where $\O_i$ is the ring of integers of K_i. The discriminant of  T also equals $[\tilde{\mathbf{T}}:\mathbf{T}]\cdot \prod_{i=1}^n\mbox{\rm disc}(K_i)$.

We view the primes dividing $\prod_{i=1}^n\mbox{\rm disc}(K_i)$ as arising from singularities of the individual irreducible components of $\mbox{\rm Spec}(\mathbf{T})$. The primes dividing $[\tilde{\mathbf{T}}:\mathbf{T}]$ 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 $S_2(\Gamma_0(389))$ is

$$2^{53}\cdot{}3^4\cdot{}5^6\cdot{}31^2\cdot{}37\cdot{}389 \cdot{}3881\cdot{}215517113148241\cdot{}477439237737571441.$$

This discriminant was computed by applying the definition of discriminant to a representation of the first 65 Hecke operators $T_1,\ldots, T_{65}$. 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), $\mathbf{T}\otimes\mathbf{Q}= K_1\times K_2\times K_3\times K_6\times K_{20},$where K_d has degree d over  Q. We have

$$\begin{eqnarray*}K_1&=&\mathbf{Q}, \\ K_2&=&\mathbf{Q}(\sqrt{2})\\ K_3&=&\math... ...delta^5 -1087\delta^4-12558\delta^3-942\delta^2+960\delta+148=0. \end{eqnarray*}$$

The discriminants of the K_i are

K1	K2	K3	K6	K20
1	23	$\quad 2^2\cdot 37$	$\quad 5^3\cdot 3881$	$\quad 2^{14}\cdot 5\cdot 389 \cdot 215517113148241\cdot 477439237737571441$

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

$$2^{19}\cdot 5^4\cdot 37\cdot 389\cdot 3881\cdot 215517113148241\cdot 477439237737571441.$$

This differs from the exact discriminant by a factor of $2^{34}\cdot 3^4\cdot 5^2\cdot 31^2$, so the index of T in its normalization is

$$[\tilde{\mathbf{T}}:\mathbf{T}]=2^{17}\cdot 3^2\cdot 5\cdot 31.$$

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.