On supersingular primes of the Elkies' elliptic curve

Abstract

Let $E$ be the elliptic curve $y^2=x^3+(i-2)x^2+x$ over the imaginary quadratic field $\mathbb{Q}(i)$. In this paper, we investigate the supersingular primes of $E$. We introduce the curve $C$ of genus two over $\mathbb{Q}$ covering a quotient of $E$ and for any prime number $p$, we state a condition (over $\mathbb{F}_p$) about the reduction of the jacobian variety of $C$ modulo $p$ which is equivalent to the existence of a supersingular prime of $E$ lying over $p$ (Theorem 5.10).