Why is $-4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2$ the discriminant of an elliptic curve?

Question

We have an elliptic curve in its normal form:

$$y^2 = f(x) = x^3 +a x^2 + bx + c,$$ where $a,b,c$ are rational numbers. The discriminant here is said to be

$$-16(4b^3 + 27c^2) \quad \text{ or } \quad 4b^3 + 27c^2.$$

However, in Rational Points on Elliptic Curves (Silverman & Tate) it is defined as

$$-4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2.$$

It looks like the first is of the same form only when $a = 0$, so the second form is more general. My question is: where on earth is this derived from? There is no such derivation given in the book.

Answer 1

Given the cubic $$ x^3 + ax^2 + bx + c, $$ replace $x$ with $X-\frac13 a$ to get $$ X^3 + BX + C = X^3 + (b-a^2/3)X + (c-ab/3+2a^3/27).$$ Now compute $4B^3 + 27C^2$ to get $$ 4b^3 + 27c^2 + 4a^3c - a^2b^2 - 18abc. $$