examples of elliptic curves with complex multiplication

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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
$\mathbb{Q}(\sqrt{-1})$	$y^{2}=x^{3}+x$
$\mathbb{Q}(\sqrt{-2})$	$y^{2}=x^{3}+4x^{2}+2x$
$\mathbb{Q}(\sqrt{-3})$	$y^{2}+y=x^{3}$
$\mathbb{Q}(\sqrt{-7})$	$y^{2}+xy=x^{3}-x^{2}-2x-1$
$\mathbb{Q}(\sqrt{-11})$	$y^{2}+y=x^{3}-x^{2}-7x+10$
$\mathbb{Q}(\sqrt{-19})$	$y^{2}+y=x^{3}-38x+90$
$\mathbb{Q}(\sqrt{-43})$	$y^{2}+y=x^{3}-860x+9707$
$\mathbb{Q}(\sqrt{-67})$	$y^{2}+y=x^{3}-7370x+243528$
$\mathbb{Q}(\sqrt{-163})$	$y^{2}+y=x^{3}-2174420x+1234136692$

Major Section:

Reference

Type of Math Object:

Example

Parent:

complex multiplication

Mathematics Subject Classification

11G05 Elliptic curves over global fields

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