Given a cubic number field and a basis $\{\gamma\_1,\gamma\_2,\gamma\_3\}$ for it over the rationals, we can write down the norm equation $N(x\_1\gamma\_1+x\_2\gamma\_2+x\_3\gamma\_3)=1$. For almost all substitutions, say $x\_1=c$, the resulting affine cubic curve is an affine part of an elliptic curve.


I was wandering what can be said of the converse. If we are given an elliptic curve over the rationals, is there a cubic number field such that for some substitution (any kind) in the norm equation, we get an affine curve isomorphic to an affine part of an elliptic curve?


I've started reading Serre's Algebraic Groups and Class Fields, which seems relevant, since its main results concern rational maps $C\rightarrow G$ from a curve to a commutative algebraic group, which is the case above.



When is an affine part of an elliptic curve isomorphic to an affine part of a norm equation?
If you have $C$ a curve of genus one and $P,Q,R$ points on it such that $P+Q \sim 2R, Q+R \sim 2P, P+R \sim 2Q$ (any two implies the third, btw), then $P-Q,R-P,Q-R$ have order three and, if you embed $C$ in the plane by the linear system $P+Q+R$ and choose coordinates in the affine plane such that the (inflectional) tangents at $P,Q,R$, which are collinear, meet at the origin, then the equation for $C$ will be of the form $f(x,y)=1$ where $f$ is a homogeneous cubic. The converse is also true. Your question has a further issue of rationality, as you want things defined over $\mathbb{Q}$. My guess is that, as long as $C$ is defined over $\mathbb{Q}$ and $P+Q+R$ is also defined over $\mathbb{Q}$ as a divisor, it should work out.


Edit: David's comment below correctly points out that not all curves of genus 1 can be put on the form $f(x,y)=1$. The mistake, as usual, is in the unjustified assertion. The inflectional tangents are not collinear. I think my method leads to an equation in the form $L\_1L\_2L\_3=1$, notation as in David's answer. Note that, in this kind of equation, the lines $L\_i=0$ are inflectional tangents at points of the cubic on the line at infinity.


