Tamagawa numbers of elliptic curves with prescribed torsion subgroup or isogeny

Abstract

We study Tamagawa numbers of elliptic curves with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over cubic fields and of elliptic curves with an $n-$isogeny over $\mathbb{Q}$, for $n\in \{6,8,10,12,14,16,17,18,19,37,43,67,163\}$. Bruin and Najman proved that every elliptic curve with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field is a base change of an elliptic curve defined over $\mathbb{Q}$. We find that Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field are always divisible by $14^2$, with each factor $14$ coming from a rational prime with split multiplicative reduction of type $I_{14k},$ one of which is always $p=2.$ The only exception is the curve 1922.e2, with $c_E=c_2=14.$ The same curves defined over cubic fields over which they have torsion subgroup $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ turn out to have the Tamagawa number divisible by $14^3$. As for $n-$isogenies, Tamagawa numbers of elliptic curves with an $18-$isogeny must be divisible by 4, while elliptic curves with an $n-$isogeny for the remaining $n$ from the mentioned set must have Tamagawa numbers divisible by 2, except for finite sets of specified curves.