= 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 $E(\Q)$ to $E(\F_p) \mod p$
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This construction taks as input:
* <elliptic curve over the rational numbers>
* a prime number $p$
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 have
$$
\frac{3}{4} \to 3 \times 4^{-1} \mod 11 = 3 \times 3 \mod 11 = 9 \mod 11
$$
where $4^{-1} = 3 \mod 11$ because $4 \times 3 = 1 \mod 11$, related: https://math.stackexchange.com/questions/1204034/elliptic-curve-reduction-modulo-p