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

TL;DR

This paper proves that for elliptic curves over quadratic fields with a point of order 13, the 13-adic valuation of their Tamagawa number is always even, confirming a conjecture and identifying a unique curve with valuation 2.

Contribution

The paper proves Krumm's conjecture that the 13-adic valuation of Tamagawa numbers is even for these elliptic curves and identifies the unique curve with valuation 2.

Findings

The 13-adic valuation of Tamagawa numbers is always even.

There exists a unique elliptic curve with valuation 2.

The conjecture by Krumm is confirmed.

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 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 $v_{13}(c_E)=2$.