A "reduced fraction" is a fraction that has the smallest possible <integer> <numerator> and <denominator> for its value.

For example:
$$
\frac{3}{6}
$$
is not a reduced fraction, because there is another fraction equal to it but with smaller <numerator> and <denominator>:
$$
\frac{1}{2}
$$


Reduced fraction

= Reduction of an elliptic curve from $E(\Q)$ to $E(\F_p) \mod p$
{synonym}
{title2}

This construction takes 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


Ciro Santilli @cirosantilli 34

 Incoming links: Denominator