On an elliptic curve defined over \(\mathbb{Q}(\sqrt -23)\) (Q2849189)

Let \(K\) be a number field and \(\mathbb A\) the adele ring of \(K\). An elliptic curve \(E\) defined over \(K\) is modular if there exists an automorphic cusp eigenform \(f\) of \(\mathrm{GL}_2(\mathbb A)\) such that the \(\mathfrak p\)-component of \(L\)-function of \(E\) is equal to that of \(L\)-function of \(f\) for any prime ideal \(\mathfrak p\) of \(K\). In the cases that \(K=\mathbb Q\) or \(K\) is a totally real number field, the modularity of elliptic curves over \(K\) have been proved by A. Wiles and others. However, in the case that \(K\) is an imaginary quadratic field, the modularity of elliptic curves without complex multiplication has been shown only for three elliptic curves defined over \(\mathbb Q(\sqrt{-23})\) in [the first author et al., Math. Comput. 79, No. 270, 1145--1170 (2010; Zbl 1227.11073)]. NEWLINENEWLINENEWLINEIn this article, the authors show that the elliptic curve \(E\) defined by the equation \(y^2+(\omega+1)xy+y=x^3+(\omega+1)x^2-7x+(5-3\omega)\), (\(\omega=(1+\sqrt{-23})/2\)) is modular, by using the algorithm to check the modularity in the article cited above. The automorphic cusp eigenform \(f\) associated with \(E\) is \(f_{44}\) in \textit{M. Lingham}'s list [Modular forms and elliptic curves over imaginary quadratic fields. Nottingham: University of Nottingham (PhD Thesis) (2005)]. In particular, \(E\) satisfies Hasse-Weil conjecture.