User allan macleod - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:01:38Z http://mathoverflow.net/feeds/user/28043 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130893/weiestrass-form/130919#130919 Answer by Allan MacLeod for Weiestrass Form Allan MacLeod 2013-05-17T07:18:13Z 2013-05-17T07:18:13Z <p>You can rewrite the form as \begin{equation*} y^2=\frac{x^2+2(2\rho+1)x+1}{x^2-2(2\rho+1)x+1} \end{equation*} so, for rational solutions (which I presume you want), there exists $z \in \mathbb{Q}$ with \begin{equation*} z^2=(x^2+2(2\rho+1)x+1)(x^2-2(\rho+1)x+1)=x^4-2(8\rho^2+8\rho+1)x^2+1 \end{equation*}</p> <p>This quartic can be transformed to an equivalent elliptic curve using the method described by Mordell in his book Diophantine Equations. We get (after some fiddling) \begin{equation*} v^2=u(u+4\rho^2+4\rho)(u+4\rho^2+4\rho+1) \end{equation*} with $x=v/u$.</p> <p>For example, $\rho=11$ gives a rank $1$ elliptic curve with point $(147,8190)$, giving $x=8190/147$ and $y=\pm 527/163$.</p> <p>This elliptic curve could be transformed to Weierstrass form if you want, but the above form is probably more useful.</p> http://mathoverflow.net/questions/112154/what-cases-say-about-the-analytic-rank-of-rank-8-elliptic-curve-457532830151317a/112168#112168 Answer by Allan MacLeod for What CASes say about the analytic rank of rank 8 elliptic curve '457532830151317a1' Allan MacLeod 2012-11-12T11:54:47Z 2012-11-12T11:54:47Z <p>Denis Simon's ellrank codes, running in Pari, give rank=8 and the following points of infinite order:</p> <p>[554, 12563], [-1103/16, 96375/64], [3553/16, 164879/64], </p> <p>[20233/144, 1090747/1728], [29592/169, 3237242/2197], [34277/169, 4653986/2197], </p> <p>[47773/361, 2532167/6859], [79874/1849, 9883308/79507]</p>