Is torsion subgroup of elliptic curve birationally invariant?

Question

It's probably a very basic question: Having two birationally equivalent elliptic curves over $\mathbb{Q}$ - is the torsion subgroup unchanged under the birational equivalence?

Answer 1

Two birationally equivalent, projective curves without singularities are isomorphic. Hence two birationally equivalent elliptic curves are isomorphic.

Answer 2

The group structure of the torsion subgroup may be the same, but the group law may look very different! I find the following example to be interesting and related to your question.

Let $E:y^2=x^3+1$ with zero at $[0,1,0]$, and consider $E':y^2=x^3+1$ where we now declare zero to be $[2,3,1]$. Then, $E$ and $E'$ are clearly birationally equivalent via the identity map but zero in $E$ does not map to zero in $E'$. Nonetheless, their torsion subgroup is both $\mathbb{Z}/6\mathbb{Z}$, but the group law is different.

For instance, let $P=[-1,0,1]$ and $Q=[0,1,0]$, then $$P+_E Q = [2,-3,1]$$ while $$P+_{E'} Q = [0,-1,0].$$