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[parent] examples of elliptic curves with complex multiplication (Example)

Here we show some elliptic curves defined over $\Rats$ 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
$\Rats(\sqrt{-1})$ $y^2=x^3+x$
$\Rats(\sqrt{-2})$ $y^2=x^3+4x^2+2x$
$\Rats(\sqrt{-3})$ $y^2+y=x^3$
$\Rats(\sqrt{-7})$ $y^2+xy=x^3-x^2-2x-1$
$\Rats(\sqrt{-11})$ $y^2+y=x^3-x^2-7x+10$
$\Rats(\sqrt{-19})$ $y^2+y=x^3-38x+90$
$\Rats(\sqrt{-43})$ $y^2+y=x^3-860x+9707$
$\Rats(\sqrt{-67})$ $y^2+y=x^3-7370x+243528$
$\Rats(\sqrt{-163})$ $y^2+y=x^3-2174420x+1234136692$



"examples of elliptic curves with complex multiplication" is owned by alozano.
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See Also: the arithmetic of elliptic curves


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Cross-references: curve, ring of integers, isomorphic, class number, quadratic imaginary field, complex multiplication, elliptic curves
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This is version 1 of examples of elliptic curves with complex multiplication, born on 2004-05-26.
Object id is 5872, canonical name is ExamplesOfEllipticCurvesWithComplexMultiplication.
Accessed 4166 times total.

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AMS MSC: 11G05 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over global fields)
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