# examples of elliptic curves with complex multiplication

Here we show some elliptic curves defined over $\mathbb{Q}$ which have complex multiplication by a quadratic imaginary field $K$ of class number $1$ (with $\operatorname{End}(E)$ exactly isomorphic to the full ring of integers $\mathcal{O}_{K}$).

$K$ Curve
$\mathbb{Q}(\sqrt{-1})$ $y^{2}=x^{3}+x$
$\mathbb{Q}(\sqrt{-2})$ $y^{2}=x^{3}+4x^{2}+2x$
$\mathbb{Q}(\sqrt{-3})$ $y^{2}+y=x^{3}$
$\mathbb{Q}(\sqrt{-7})$ $y^{2}+xy=x^{3}-x^{2}-2x-1$
$\mathbb{Q}(\sqrt{-11})$ $y^{2}+y=x^{3}-x^{2}-7x+10$
$\mathbb{Q}(\sqrt{-19})$ $y^{2}+y=x^{3}-38x+90$
$\mathbb{Q}(\sqrt{-43})$ $y^{2}+y=x^{3}-860x+9707$
$\mathbb{Q}(\sqrt{-67})$ $y^{2}+y=x^{3}-7370x+243528$
$\mathbb{Q}(\sqrt{-163})$ $y^{2}+y=x^{3}-2174420x+1234136692$
Title examples of elliptic curves with complex multiplication ExamplesOfEllipticCurvesWithComplexMultiplication 2013-03-22 14:22:47 2013-03-22 14:22:47 alozano (2414) alozano (2414) 4 alozano (2414) Example msc 11G05 ArithmeticOfEllipticCurves