Weierstrass equation of an elliptic curve
Moreover, the following proposition specifies any possible change of variables.
Let be an elliptic curve given by a Weierstrass model of the form:
with . Then:
The only change of variables preserving the projective point and which also result in a Weierstrass equation, are of the form:
with and .
Any two Weierstrass equations for differ by a change of variables of the form given in .
Once we have one Weierstrass model for a given elliptic curve , and as long as the characteristic of is not or , there exists a change of variables (of the form given in the previous proposition) which simplifies the model considerably.
Let be a field of characteristic different from or . Let be an elliptic curve defined over . Then there exists a Weierstrass model for of the form:
where are elements of .
Let be an elliptic curve and let
be two distinct Weierstrass models for . Then (by Prop. 1) there exists a change of variables of the form:
with and . Moreover, , i.e. the invariants are equal ( is defined in http://planetmath.org/node/JInvariantthis entry) and , where is the discriminant (as defined in http://planetmath.org/node/JInvarianthere).
|Title||Weierstrass equation of an elliptic curve|
|Date of creation||2013-03-22 15:48:00|
|Last modified on||2013-03-22 15:48:00|
|Last modified by||alozano (2414)|