The j invariant of an elliptic curve - Maple Help

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algcurves[j_invariant] - The j invariant of an elliptic curve

Calling Sequence

j_invariant(f, x, y)

Parameters

f

-

polynomial in x and y representing a curve of genus 1

x, y

-

variables

Description

• 

For algebraic curves with genus 1 one can compute a number called the j invariant. An important property of this j invariant is the following: two elliptic (i.e. genus = 1) curves are birationally equivalent (i.e. can be transformed to each other with rational transformations over an algebraically closed field of constants) if and only if their j invariants are the same.

• 

The curve must be irreducible and have genus 1, otherwise the j invariant is not defined and this procedure will fail.

Examples

withalgcurves:

f:=y5+4323y23+11y317y4316x23+16x334x43:

Check that the genus is 1, because only then is the j invariant defined.

genusf,x,y

1

(1)

j_invariantf,x,y

1404928171

(2)

See Also

algcurves[genus], algcurves[Weierstrassform]


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