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تسجيل مستخدم جديدLocal data of rational elliptic curves with non-trivial torsion
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By Mazurs Torsion Theorem, there are fourteen possibilities for the non-trivial torsion subgroup $T$ of a rational elliptic curve. For each $T$, we consider a parameterized family $E_T$ of elliptic curves with the property that they parameterize all elliptic curves $E/mathbb{Q}$ which contain $T$ in their torsion subgroup. Using these parameterized families, we explicitly classify the N{e}ron type, the conductor exponent, and the local Tamagawa number at each prime $p$ where $E/mathbb{Q}$ has additive reduction. As a consequence, we find all rational elliptic curves with a $2$-torsion or a $3$-torsion point that have global Tamagawa number~$1$.
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