Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p (Reduction of an elliptic curve from $E(\mathbb{Q})$ to $E({\mathbb{F}}_p)\mathrm{m}\mathrm{o}\mathrm{d}p$)

This construction takes as input:and it produces an elliptic curve over a finite field of order $p$ as output.

The constructions is used in the Birch and Swinnerton-Dyer conjecture.

To do it, we just convert the coefficients $a$ and $b$ from the Equation "Definition of the elliptic curves" from rational numbers to elements of the finite field.

For example, suppose we have $a=3/4$ and we are using $p=11$.

For the denominator $4$, we just use the multiplicative inverse, e.g. supposing we havewhere $4^{-1}=3\mathrm{m}\mathrm{o}\mathrm{d}11$ because $4\times 3=1\mathrm{m}\mathrm{o}\mathrm{d}11$, related: math.stackexchange.com/questions/1204034/elliptic-curve-reduction-modulo-p