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For each positive integer $n$, the congruent number curve $E_n$ is the elliptic curve defined over $\Q$ with equation $y^2 = x(x^2-n^2)$. Rational points $(x,y)\in E_n(\Q)$ with $y\not=0$ are associated to solutions of the congruent number problem for $n$: given rational numbers $X,Y,Z$ satisfying $X^2+Y^2=Z^2$ and $XY=2n$, set $x=\frac{1}{4}Z^2$ and $y=\frac{1}{8}Z(X^2-Y^2)$; then $y^2=x(x^2-n^2)$.

The only rational points of finite order on $E_n(\Q)$ are the point at infinity (the origin for the group law) and the three points of order $2$: $(0,0)$, $(\pm n,0)$; thus, $n$ is congruent if and only if $E_n(\Q)$ is infinite, that is, if and only if the rank of $E_n(\Q)$ is positive.

For example, the elliptic curve $E_6: y^2 = x^3 - 36x$ has rank $1$, with Mordell-Weil group $E_6(\Q)$ generated (modulo torsion) by $P = (-3,9)$, and $2P=(25/4,-35/8)$ is the rational point arising from the $(3,4,5)$-triangle. In general, only points in $2E_n(\Q)$ come from solutions to the congruent number problem; these are the rational points $(x,y)$ for which $x$ and $x\pm n$ are all squares.

The conductor of $E_n$ is $32n_0^2$ when the square-free part $n_0$ of $n$ is odd, and $16n_0^2$ when $n_0$ is even.

Each congruent number elliptic curve $E_n$ belongs to an isogeny class of four isogenous curves, connected by 2-isogenies. The curves $2$-isogenous to $E_n=[0,0,0,-n^2,0]$ are $[0,0,0,4n^2,0]$ and $[0,\pm 12n,0,n^2,0]$.

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  • Last edited by John Cremona on 2021-07-14 09:40:20
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