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?


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).


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