Rouse classification (reviewed)

J. Rouse and D. Zureick-Brown have classified all possible images $G$ of the 2-adic Galois representation attached to an elliptic curve defined over $\Q$, in terms of the index $n$ of $G$ in $\GL(2,\Z_2)$, the level $l=2^k$, and explicit generators of $G$.

The Rouse level $l=2^k$ is the smallest power of $2$ such that $G$ contains the kernel of reduction modulo $l$. The exponent $k$ is called the Rouse log-level. The Rouse index of $G$ is the index $[\GL(2,\Z_2):G]$.

The Rouse generators of $G$ are a finite set of matrices in $\GL(2,\Z/l\Z)$ whose pull-backs to $\GL(2,\Z_2)$ generate $G$.