our statement comes from the fact that we do not expect forms of minimal level to always lift to characteristic zero: this is why the statement of Conjecture \[conj:loweringlevel\] is in terms of group cohomology over $\F_{\ell}$.

The second section contains the main theorem, and its proof when the image is absolutely irreducible. The third section studies elliptic curves over $K$ with complex multiplication. The main results includes a complete classification of all CM curves of odd prime power discriminant. The case of small image at an odd prime is treated in the fourth section, where elliptic curves over $K$ of odd prime power conductor with a rational point of odd order $\ell$ are considered. In Theorem \[thm:3-torsionspecialcase\] we prove that if $\ell=3$ and $K=\Q(\sqrt{-3})$, then any such curve either has discriminant valuation not divisible by $3$, or its $3$-isogenous curve satisfies this property. For all other choices of $\ell$ and $K$, we give a finite list of possible curves: see Table \[table:oddtorsioncurves\].

The fifth section studies curves of odd conductor over $K$ with a rational $2$-torsion point. An important result is Theorem \[thm:classification2torsion\], where a description of all such curves (up to twist) is given. This result is very 