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  "body": "\n## Summary\n\nThe Selmer group of an isogeny of [[elliptic curves|427N6S6QQW]] over a [[number field|MWBDFS2KJ5]] is a group of [[Galois cohomology|9MPCZ3EAJM]] classes.\nThese classes satisfy local conditions at every place, and the group is finite and computable.\nIt contains a copy of the quotient $E'(K)/\\phi(E(K))$, and the difference is the $\\phi$-torsion of the [[Tate–Shafarevich group|BK9WBMF56G]].\nComputing a Selmer group is the standard way to bound the rank of $E(K)$ from above.\nIt is the effective core of the proof of [[Mordell's theorem|BZHGKN6Q3K]].\n\n## Explanations\n\n### L1 human\n\nTo find all rational points of an elliptic curve, one would like to describe the points modulo doubles, $E(\\Q)/2E(\\Q)$.\nThis group sits inside a larger group of candidates that one can compute from local information, prime by prime.\nThat larger group is the $2$-Selmer group.\nIf it is as small as the known points allow, the rank is determined.\nIf it is larger, the extra candidates come from points not yet found or from elements of the Tate–Shafarevich group.\n",
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