TITLE: The ring of algebraic integers of the number field generated by torsion points on an elliptic curve
QUESTION [15 upvotes]: (Warning: a student asking)
Let $E$ be an elliptic curve over $\mathbf Q$. Let $P(a,b)$ be a (nontrivial) torsion point on $E$. Is there an easy description of the ring of algebraic integers of $\mathbf Q(a,b)$? I'm curious about the answer for general elliptic curves, but I'm not sure whether such an answer is possible.
(This question is motivated by the nice description of the rings of integers of cyclotomic fields $\mathbf Q(\zeta_n)$)

REPLY [4 votes]: For further work on monogeneity questions, you might want to have a look at
some of the papers of Reinhard Schertz (I'm afraid that I don't have precise
references right now). He apparently also has a new book out entitled "Complex
Multiplication" or something like this, which I've not seen, but which probably
also discusses some of his work on this topic.