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# Mathematical Research Letters

## Volume 10 (2003)

### Number 2

### Icosahedral $\mathbb Q$-Curve Extensions

Pages: 205 – 217

DOI: http://dx.doi.org/10.4310/MRL.2003.v10.n2.a8

#### Author

#### Abstract

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.