TAMAGAWA NUMBERS OF ELLIPTIC CURVES WITH C13 TORSION OVER QUADRATIC FIELDS.

Let E be an elliptic curve over a number field K, c_v the Tamagawa number of E at v, and let c_E = ∏_v c_v. Lorenzini proved that v13(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 cE 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 v13(c_E)=2. [ABSTRACT FROM AUTHOR]