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 $\mathrm{End}(E)$ exactly isomorphic to the full ring of integers ${𝒪}_K$).

$K$	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
Canonical name	ExamplesOfEllipticCurvesWithComplexMultiplication
Date of creation	2013-03-22 14:22:47
Last modified on	2013-03-22 14:22:47
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Example
Classification	msc 11G05
Related topic	ArithmeticOfEllipticCurves