English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

By Mazur's Torsion Theorem, there are fourteen possibilities for thenon-trivial torsion subgroup $T$ of a rational elliptic curve. For each $T$,such that $E$ may have additive reduction at a prime $p$, we consider aparameterized family $E_T$ of elliptic curves with the property that theyparameterize all elliptic curves $E/\mathbb{Q}$ which contain $T$ in theirtorsion subgroup. Using these parameterized families, we explicitly classifythe Kodaira-N\'{e}ron type, the conductor exponent, and the local Tamagawanumber at each prime $p$ where $E/\mathbb{Q}$ has additive reduction. As aconsequence, we find all rational elliptic curves with a $2$-torsion or a$3$-torsion point that have global Tamagawa number $1$.

Local data of rational elliptic curves with non-trivial torsion