Tamagawa numbers of elliptic curves with prescribed torsion subgroup or isogeny

We study Tamagawa numbers of elliptic curves with torsion Z/2Z circle plus Z/14Z over cubic fields and of elliptic curves with an n-isogeny over Q, for n is an element of {6, 8, 10, 12, 14, 16, 17, 18, 19, 37, 43, 67, 163}. Bruin and Najman [3] proved that every elliptic curve with torsion Z/2Z circle plus Z/14Z over a cubic field is a base change of an elliptic curve defined over Q. We find that Tamagawa numbers of elliptic curves defined over Q with torsion Z/2Z circle plus Z/14Z over a cubic field are always divisible by 142, with each factor 14 coming from a rational prime with split multiplicative reduction of type I14k, one of which is always p = 2. The only exception is the curve 1922.e2, with cE = c2 = 14. The same curves defined over cubic fields over which they have torsion subgroup Z/2Z circle plus Z/14Z 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. (c) 2021 Elsevier Inc. All rights reserved.