# Discriminant of an elliptic curve

I have found different discriminants for general Weierstrass elliptic curves: $y^2 = x^3 + ax + b$ For example, WolframAlpha state it is $-16(4a^3 + 27b^2)$ on their site http://mathworld.wolfram.com/EllipticDiscriminant.html However, in books I have read like Washington's Elliptic Curves: Number Theory and Cryptography the discriminant is stated to be $-(4a^3 + 27b^2)$.

Everywhere I look seems to have either of these options, but no one gives a reason for using it. Is there one correct answer or are they both correct? If this is so, why?

## 1 Answer

You know the Weirstrass or canonical form of an elliptic curve is a trinomial $x^3+ax+b$ and a cubic curve is elliptic iff its discriminant is non-zero. For this it is not matter if this discriminant has one of the two forms you mention or even if the discriminant is $4A^3+27B^2$ (Cassels). The factor $-16$ is purely convenient for some purposes but it is irrelevant indeed and it becomes from distinct birrational transformations leading a general expression of an elliptic cubic to its canonical form as a trinomial.

Furthermore the Weierstrass function $\wp(z)$ is related to its derivative by the fundamental equation $$[\wp’(z)]^2=4[\wp(z)]^3+A\wp(z)+B$$ for which some authors use the canonical form $4x^3+ax+b$ instead of $x^3+ax+b$.

On the other hand, in Number Theory, the discriminant of a trinomial $x^n+ax+b$ (think Weierstrass for $n=3$) is defined by $$D(1,x,…x^{n-1})= (-1)^{\frac{n(n-1)}{2}}[n^nb^{n-1}+(-1){n-1}(n-1)^{n-1}a^n]$$ hence for the trinomial $x^3+ax+b$ you get $$D(1,x,x^2)=(-1)^3[27b^2+4a^3]=-(4a^3+27b^2)$$ in accordance with Washington.

Concerning Wolfram, you can see in the first step of his simplification towards the canonical form he gets, the expression $$y^2=4x^3+b_2x^2+2b_4x+b_6$$ and a “simpler” discriminant for “another” polynomial; he writes “The discriminant depends on the choice of equations, and can change after a change of variables” but he is careful to add “unlike the j-invariant

Look at the coefficient $4$ of $x^3$ and note the subsequent step of change of variables trying to see that he could avoid if he have wished the elimination of this coefficient.

That´s all with my deficient English (I hope it can help you something).