Some applications of the traces of Frobenius of elliptic curves in a certain family

Abstract

Let J = {a, b} be an unordered pair of elements of Fq and EJ the associated elliptic curve of the form y 3 = (x − a)(x − b) over Fq. Using its trace of Frobenius, we obtain several applications. We first compute the average analytic rank of elliptic curves for our family and generate elliptic curves with designated extremal primes. Furthermore, we determine explicit and average values on class numbers of every constant field extension of KJ := Fq(T, p3 (T − a)(T − b)). Finally, we estimate the exact values and average values on Euler-Kronecker constants of KJ .