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  "body": "\n## Statement\n\nLet $a, b \\in \\Z$ with $D = 4a^3 + 27b^2 \\ne 0$, and let $E$ be the [[elliptic curve|427N6S6QQW]] $y^2 = x^3 + ax + b$ over $\\Q$.\nLet $P = (x, y) \\in E(\\Q)$ be a point of finite [[order|0WE2NHZJ7B]] in the [[group of rational points|P7V1BQJPSS]], with $P \\ne O$.\nThen $x$ and $y$ are integers, and either $y = 0$ or $y^2$ divides $D$.\n\n## Remarks\n\nThe integrality comes from a study of the denominators of points in the $p$-adic filtration: points whose coordinates have $p$ in the denominator form a torsion-free subgroup.\nDivisibility follows next: if $P$ has finite order, so does $2P$, and both have integer coordinates; the duplication formula then forces $y^2 \\mid D$.\nThe theorem gives an algorithm for the torsion subgroup: list the finitely many candidates and test each for finite order.\nExample: on $y^2 = x^3 + 1$ one has $D = 27$, so a torsion point with $y \\ne 0$ has $y \\in \\set{\\pm 1, \\pm 3}$.\nThis gives the points $(0, \\pm 1)$ and $(2, \\pm 3)$.\nWith $(-1, 0)$ and $O$ they form a cyclic group of order $6$.\nThe converse fails: on $y^2 = x^3 + 2$ the point $(-1, 1)$ has integer coordinates and $1 \\mid 108$, but it has infinite order.\nThe hypothesis that $a$ and $b$ are integers is needed; any curve can be scaled to this form by $(x, y) \\mapsto (u^2 x, u^3 y)$.\n\n## Explanations\n\n### L1 human\n\nA rational point of finite order on such a curve is very special.\nIts coordinates are whole numbers, and the $y$-coordinate is zero or has a square that divides a fixed number.\nSo the points of finite order can be found by a finite search.\n\n### L2 human\n\nThe theorem was found independently by Nagell and Lutz in the 1930s.\nIt is the standard tool, together with reduction modulo good primes, to compute the torsion subgroup $T$ of $E(\\Q)$ in [[Mordell's theorem|BZHGKN6Q3K]].\n[[Mazur's torsion theorem|RQS6YHAR8N]] then lists all groups that can occur.\n",
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