Tamagawa Numbers of elliptic curves with 𝐶₁₃ torsion over quadratic fields

Abstract

Let E E be an elliptic curve over a number field K K , c v c_v the Tamagawa number of E E at v v , and let c E = ∏ v c v c_E=\prod _{v}c_v . Lorenzini proved that v 13 ( c E ) v_{13}(c_E) is positive for all elliptic curves over quadratic fields with a point of order 13 13 . Krumm conjectured, based on extensive computation, that the 13 13 -adic valuation of c E 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 v_{13}(c_E)=2 .