While reading "Rational Points on Elliptic Curves" by Silverman and Tate, I came across this interesting passage about Bachet's duplication formula:
I know how to derive Bachet's duplication formula using the tangent method (namely, one constructs a tangent line to a given rational point on this cubic. The third intersection point of the line and the cubic gives the new rational point). My question is about the mysterious $-432$.
Why is it true that if the original rational solution $(x, y)$ has $xy\neq 0$ and if
$c\neq 1, -432$, then repeating this process leads to infinitely many
distinct solutions?
Edit. Later in the book (page 24), there is a explicit calculation that puts the cubic $u^3+v^3=\alpha$ into Weierstrass Normal Form. The resulting equation is $y^2=x^3-432\alpha^2$. So in particular, $y^2=x^3-432$ is Weierstrass Normal Form of $x^3+y^3=1$. So perhaps this explains the appearance of $-432$ above?
Best Answer
We need the convention that $c$ is not divisible by a $6$th power, since for any nonzero integer $m$ the equations $y^2=x^3+c$ and $y^2 =x^3 +m^6 c$ are easily transformed into each other by a rational change of variables, so their rational solutions are in a simple bijection. It's for sixth-power free integers $c$ that $y^2 =x^3 +c$ has torsion points with nonzero $x$ and $y$ only when $c=1$ or $−432$. In any event, I think this is discussed in the little book by Cassels on elliptic curves. It's mentioned as Prop. 17.10.1 in Ireland & Rosen. See also Prop 17.9.1 in Ireland & Rosen.