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Abstract
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By Mazur’s torsion theorem, there are fourteen possibilities for the nontrivial torsion subgroup
of a rational elliptic
curve. For each
, such that
may have additive reduction
at a prime
, we consider
a parametrized family
of elliptic curves with the property that they parametrize all elliptic curves
which
contain
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
where
has additive reduction. As a consequence, we find all rational elliptic curves with a
- or
-torsion point that have
global Tamagawa number .
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Keywords
elliptic curves, Tamagawa numbers, Kodaira–Néron-types,
Tate's algorithm
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Mathematical Subject Classification
Primary: 11G05, 11G07, 11G40, 14H52
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Milestones
Received: 11 December 2020
Revised: 14 November 2021
Accepted: 5 March 2022
Published: 1 August 2022
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