2021/02/09 by Trbović, Antonela · 1 citation
#11G05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2102.04834
We study Tamagawa numbers of elliptic curves with torsion ℤ/2ℤ⊕ ℤ/14ℤ over cubic fields and of elliptic curves with an n-isogeny over ℚ, for n∈\6,8,10,12,14,16,17,18,19,37,43,67,163\. Bruin and Najman proved that every elliptic curve with torsion ℤ/2ℤ⊕ 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⊕ 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⊕ ℤ/14ℤ turn out to have the Tamagawa number divisible by 143. 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.