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There have been many computational investigations into modularity of
elliptic curves over number fields other than Q, especially for
imaginary quadratic fields (Cremona and his students,
Dieulefait--Guerberoff--Pacetti) and real quadratic fields (Dembele).
In this talk we present joint work in progress with F. Hajir,
D. Ramakrishnan, and D. Yasaki that treats the totally complex quartic
field of fifth roots of unity. We will discuss our computational
techniques and will give examples of elliptic curves over this field
whose L-series apparently match those given by the Hecke data of
eigenforms.
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