Icosahedral $\mathbb Q$-Curve Extensions

We consider elliptic curves defined over $\mathbb Q(\sqrt{5})$ which are either 2- or 3-isogenous to their Galois conjugate and which have an absolutely irreducible mod 5 representation. Using Klein’s classical formulas which associate an icosahedral Galois extension $K/\mathbb Q$ with the 5-torsion of an elliptic curve, we prove that there is an association of such extensions generated by quintics $x^5 + A \, x^2 + B \, x + C$ satisfying $A \, B = 0$ with the aforementioned elliptic curves.

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Published 1 January 2003