Keywords: 389

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# 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

where . 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

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- Rank
- Cremona's data
- Characteristic polynomials