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  "body": "\n## Statement\n\nLet $E$ be an [[elliptic curve|427N6S6QQW]] over $\\Q$, and let $E(\\Q)_{\\mathrm{tors}}$ be the [[torsion subgroup|0DBWXQRXQJ]] of the [[group of rational points|P7V1BQJPSS]].\nThen $E(\\Q)_{\\mathrm{tors}}$ is isomorphic to one of the following $15$ groups:\n\n1. the [[cyclic group|0KVGH0J5B3]] $\\Z/N\\Z$ with $1 \\le N \\le 10$ or $N = 12$;\n2. $\\Z/2\\Z \\times \\Z/2N\\Z$ with $1 \\le N \\le 4$.\n\n## Remarks\n\nEach of the $15$ groups occurs for infinitely many curves with different $j$-invariants; this part was known before Mazur, from explicit families.\nThe hard part is that no other group occurs.\nA rational point of order $N$ on $E$ gives a non-cuspidal rational point on the modular curve $X_1(N)$.\nMazur proved in 1977, with his study of the Eisenstein ideal, that for primes $N \u003e 7$ the curve $X_1(N)$ has no such points.\nTogether with earlier results on the composite orders, this gives the list.\nExample: on $y^2 = x^3 + 1$ the torsion subgroup is cyclic of order $6$.\nOver larger number fields, the list of possible torsion groups is different, but it is still finite for each degree, by a theorem of Merel.\n\n## Explanations\n\n### L1 human\n\nThe points of finite order on an elliptic curve over the rationals form a small finite group.\nMazur's theorem gives the complete list of what that group can be: never more than $16$ points, and only $15$ shapes.\n\n### L2 human\n\nTogether with the [[Nagell–Lutz theorem|G38B7E4ESW]], it makes the torsion part of [[Mordell's theorem|BZHGKN6Q3K]] fully explicit.\nThe method, through rational points on modular curves, is an early deep use of modular curves in arithmetic.\n",
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