In Darmon's paper on p.14 he lists a table of signatures $(p,q,r)$ and constructed Frey curves. How do you construct the Frey curve he gives for $(2,3,p)$?
The curve he gives for this signature is:
$y^2=x^3+3bx+2a$
The Frey curve that Poonen,Schaefer,and Stoll give on p.9 of their paper for the signature $(2,3,7)$ is:
$y^2=x^3+3bx-2a$
My construction question extends to their curve also. How are these curves constructed?
On
Apparently, I asked a question that was asked six years ago on Math OverFlow, and answered by Noam Elkies. He said,
"Factor $x^3+ax+b=(x−x1)(x−x2)(x−x3)$. Translate $x$ by $x1$ to make the first factor $x$. Then multiply $x$ by $x2−x1$ and $y$ by $(x2−x1)^{3/2}$ and divide by $(x2−x1)^3$ to get $y^2=x(x−1)(x−L)$ with $L=(x3−x1)/(x2−x1)$. If $x^3+ax+b$ does not factor completely then $L$ must be in a field larger than the one used to define the curve. – Noam D. Elkies Jun 19 '12 at 4:28"
Also, this post seems to be relevant as well.
Assuming that $a^2+b^3=c^p$, the discriminant of the elliptic curve $$ E \; \; : \; \; y^2=x^3+3bx \pm 2a $$ is just $$ -1728 (b^3+a^2) = -1728 c^p. $$ Hence, for suitably large prime $p$, we have, after level lowering, a correspondence between $E$ and a particular weight $2$ modular form of level dividing $1728$. The key point here is that $E$ has discriminant that is a $p$-th power outside of $2$ and $3$ (similar to those curves used for equations like $a^p+b^p=c^p$, $a^p+b^p=c^2$ and $a^p+b^p=c^3$).