Discriminant of Elliptic curve $y^2=ax^3+bx^2+cx+d$?

Question

I know the discriminant of the elliptic curve of the form $y^2=x^3+px^2+q x+r$ but how to calculate the discriminant if we have $y^2=ax^3+bx^2+cx+d$?

Answer 1

Your question seems to be "when does the equation $y^2 = a x^3 + b x^2 + c x +d$ define an elliptic curve"? Implicit I suppose is that you mean the corresponding projective curve for this affine model. The answer is precisely (assuming we are over a field of characteristic different $2$) when $a \ne 0$ and the RHS does not have any repeated roots, which should be the answer you are looking for. Assuming $a \ne 0$ (and is thus a unit) however, the cubic $f(x)$ has distinct roots if and only if $f(x)/a$ has distinct roots, since clearly they have the same roots. So you can then take the discriminant of this polynomial. Since you have to assume that $a \ne 0$, you may as well multiply the corresponding expression by a power of $a$ to be integral again, and so explicitly you get an elliptic curve if and only if

$$a (-b^2 c^2 + 4 a c^3 + 4 b^3 d - 18 a b c d + 27 a^2 d^2) \ne 0.$$

Warning: It is sloppy to talk about the the discriminant of an elliptic curve, which isn't really well defined. The discriminant is defined on a pair $(E,\omega)$ for some differential $\omega$ which generates $H^1(E,\Omega^1)$. In particular, if your elliptic curve is given in Weierstrass form, then it is traditional to take $\omega = dx/(2y)$, in which case $\Delta = \Delta(E,\omega)$ is $16$ times the discriminant of the monic cubic.