What can one say about the order of the torsion group of an elliptic curve defined over the compositum of all quadratic extensions of $\mathbb{Q}$ ?



Order of torsion group
I think the papers you should have a look at are those two by Fujita, great material and very well-written (in my humble opinion):


1) Y. Fujita, Torsion subgroups of elliptic curves with non-cyclic torsion over ${\mathbb Q}$ in elementary abelian 2-extensions of ${\mathbb Q}$, Acta Arith. 115 (2004) 29–45. MR2102804 (2005j:11041)


2) Fujita, Y.: Torsion subgroups of elliptic curves in elementary abelian $2$-extensions of ${\mathbb Q}$. J. Number Theory, 114, 124–134 (2005) MR2163908 (2006h:11055)


Hope this helps.


