Tamagawa Numbers of elliptic curves with $C_{13}$ torsion over quadratic fields

Abstract:

Let $E$ be an elliptic curve over a number field $K$, $c_v$ the Tamagawa number of $E$ at $v$, and let $c_E=\prod _{v}c_v$. Lorenzini proved that $v_{13}(c_E)$ is positive for all elliptic curves over quadratic fields with a point of order $13$. Krumm conjectured, based on extensive computation, that the $13$-adic valuation of $c_E$ is even for all such elliptic curves. In this note we prove this conjecture and furthermore prove that there is a unique such curve satisfying $v_{13}(c_E)=2$.