Determination of elliptic curves with everywhere good reduction over real quadratic fields (Q1305327)

Let \(K={\mathbb Q}(\sqrt{29})\), let \(\varepsilon=(5+\sqrt{29})/2\) be a fundamental unit of \(K\), let \(E\) be the elliptic curve defined by the equation \(y^2+xy+\varepsilon^2y=x^3\). \textit{T. Nakamura} [J. Math. Soc. Japan 36, 701-707 (1984; Zbl 0546.14022)] proved that any elliptic curve \(\tilde{E}\) defined over \(K\) and having everywhere good reduction over \(K\), is in fact \(K\)-isogenous to \(E\). In this text the author determines explicitly one representative for each class of \(K\)-isomorphism of elliptic curves which are \(K\)-isogenous to \(E\): there are in this way exactly \(4\) of these representatives and they have everywhere good reduction over \(K\). The phenomenon above is in accordance with a conjecture of Pinch predicting the existence of a correspondence between elliptic curves having everywhere good reduction over real quadratic number fields, and the elliptic curves associated to some \({\mathbb Q}\)-simple factors of spaces of cusp forms of nebentype and weight \(2\), constructed by \textit{G. Shimura} [Introduction to the arithmetic theory of automorphic functions. Princeton Univ. Press (1994; Zbl 0872.11023)].