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 postive 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 furhtermore prove that there is an unique such curve satisfying $v_{ 13}(c_E)=2$.
Categories: math.NT