show · ec.torsion_subgroup all knowls · up · search:

For an elliptic curve $E$ over a field $K,$ the torsion subgroup of $E$ over $K$ is the subgroup $E(K)_{\text{tor}}$ of the Mordell-Weil group $E(K)$ consisting of points of finite order. For a number field $K$ this is always a finite group, since by the Mordell-Weil Theorem $E(K)$ is finitely generated.

The torsion subgroup is always either cyclic or a product of two cyclic groups. The torsion structure is the list of invariants of the group:

  • $[]$ for the trivial group;
  • $[n]$ for a cyclic group of order $n>1$;
  • $[n_1,n_2]$ with $n_1\mid n_2$ for a product of non-trivial cyclic groups of orders $n_1$ and $n_2$.

For $K=\Q$ the possible torsion structures are $[n]$ for $n\le10$ and $n=12$, and $[2,2n]$ for $n=1,2,3,4$.

Knowl status:
  • Review status: reviewed
  • Last edited by John Cremona on 2019-02-08 11:31:12
Referred to by:
History: (expand/hide all)