Quadratic points on X0(163)

Abstract

We determine all the quadratic points on the genus 13 modular curve $X_0\left(163\right)$, thus completing the answer to a recent question of Banwait, the second-named author, and Padurariu. In doing so, we investigate a curious phenomenon involving a cubic point with complex multiplication on the curve $X_0\left(163\right)$. This cubic point prevents us, due to computational restraints, from directly applying the state-of-the-art Atkin–Lehner sieve for computing quadratic points on modular curves $X_0\left(N\right)$. To overcome this issue, we introduce a technique which allows us to work with the Jacobian of curves modulo primes by directly computing linear equivalence relations between divisors.