Elliptic curves and primes

Question

I have a question related to elliptic curves and primes:

For which primes $p \geq 5$ does the equation

$$y^2 = x^3 + 7x + 3 $$

define an elliptic curve over $F_p$?

How should I approach this problem?

Answer 1

Let $K$ be a field. An elliptic curve $E$ over $K$ is a nonsingular projective curve of genus one, equipped with a fixed $K$-rational point $O$. With this definition, your curve is technically never an elliptic curve, as you have written down an affine curve and not given a chosen point!

But let's ignore that issue. It is common to present elliptic curves in the form you've given, and your curve in particular is an affine patch of the projective curve $$ Y^2Z = X^3 + 7XZ^2 + 3Z^3. $$ Here, we typically choose the point $O$ to be the "point at infinity," which has projective coordinates $[0:1:0].$

So now we must address when the curve you've given is nonsingular. It is a well-known fact that the curve $$ y^2 = x^3 + ax + b $$ is nonsingular if and only if the discriminant $\Delta = −16(4a^3 + 27b^2)$ is nonzero. See, for example [1, Proposition III.1.4], but be aware that there is a typo in the statement: (i) in part (a) should read "It is nonsingular if and only if $\Delta\neq0$."

Now we can address the question: for your curve, we have $$ \Delta = -16(4\cdot 7^3 + 27\cdot 3^2) = -25840 = -2^4\cdot 5\cdot 17\cdot 19. $$ This will be nonzero in $\Bbb{F}_p$ if and only if $p\neq 2,5,17,19.$ So, the curve you've given is not an elliptic curve if $p = 2,5,17,$ or $19.$

Finally, you might ask what is the genus of this curve? Well, we may calculate this in the non-singular case using the genus-degree formula, which states that for an irreducible nonsingular plane curve of degree $d,$ the genus $g$ is $$ g = \frac{(d - 1)(d - 2)}{2}. $$

The curve you've written down has degree $d = 3,$ so $g = 1.$

Thus, we may conclude that $y^2 = x^2 + 7x + 3$ defines an Elliptic curve over $\Bbb{F}_p$ when $p\neq 2,5,17,19.$

[1] Silverman, Joseph. (2009). The Arithmetic of Elliptic Curves.