Elliptic modularity for octahedral Galois representations

Let $\rho:\Gal(\overline{{\bold Q}}/{\bold Q}) \to \operatorname{{\bf GL}}_2({\cosf}_3)$ be a residual representation with cyclotomic determinant and splitting field of degree at least 16. First we show that $\rho$ is isomorphic to the Galois representation attached to the 3-torsion points of an elliptic curve which is modular. Then we discuss some facts about minimality of conductors.

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Published 1 January 1996