#### Vol. 318, No. 1, 2022

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Local data of rational elliptic curves with nontrivial torsion

### Alexander J. Barrios and Manami Roy

Vol. 318 (2022), No. 1, 1–42
DOI: 10.2140/pjm.2022.318.1
##### Abstract

By Mazur’s torsion theorem, there are fourteen possibilities for the nontrivial torsion subgroup $T$ of a rational elliptic curve. For each $T$, such that $E$ may have additive reduction at a prime $p$, we consider a parametrized family ${E}_{T}$ of elliptic curves with the property that they parametrize all elliptic curves $E∕ℚ$ which contain $T$ in their torsion subgroup. Using these parametrized families, we explicitly classify the Kodaira–Néron-type, the conductor exponent and the local Tamagawa number at each prime $p$ where $E∕ℚ$ has additive reduction. As a consequence, we find all rational elliptic curves with a $2$- or $3$-torsion point that have global Tamagawa number $1$.

##### Keywords
elliptic curves, Tamagawa numbers, Kodaira–Néron-types, Tate's algorithm
##### Mathematical Subject Classification
Primary: 11G05, 11G07, 11G40, 14H52
##### Supplementary material

Kodaira--N'{e}ron-type and conductor exponent