Introduction

The Jacobian of the modular curve X₀(389) has many remarkable properties. In this note, we describe the isogeny and endomorphism structure of its Jacobian and compute the discriminant of the corresponding Hecke algebra. This work was inspired by an enquiry of K. Ribet: are there primes p such that the discriminant of the Hecke algebra associated to X₀(p) is divisible by p? The answer is yes, but the only known p<14000 with this property is p=389. It is unkown whether or not there are any other examples.

The modular curve X=X₀(389) has genus g=32, and its Jacobian J=J₀(389)is isogenous to a product of Q-simple abelian varieties

$$J \sim A_1\times A_2 \times A_3 \times A_6 \times A_{20},$$

where $\dim A_d = d$. The elliptic curve A₁ is the first curve in Cremona's tables having rank 2; it is labeled 389A. The L-function corresponding to A₂₀ does not vanish at s=1, so A₂₀ has analytic rank 0, and hence algebraic rank 0, by the theorem of Kolyvagin and Logachev. The genus of X⁺=X/w₃₈₉ is g⁺ =11, and J₀(N)^- decomposes up to isogeny as a product $J_0(N)^{-}\sim A_{1} \times A_{20}.$

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