### Abstract

We construct a parametric family {E^{(±)} (s, t, u)} of minimal Q-curves of degree 5 over the quadratic fields Q(√s^{2} + st - t^{2}), and the family {C(s, t, u)} of genus two curves over Q covering E^{(+)} (s, t, u) whose jacobians are abelian surfaces of GL_{2}-type. We also discuss the modularity for them and the sign change between E^{(+)} (s, t, u) and its twist E^{(-)} (s. t, u), which correspond by modularity to cusp forms of trivial and non-trivial Neben type characters, respectively. We find in {C (s, t, u)} concrete equations of curves over Q whose jacobians are isogenous over cyclic quartic fields to Shimura's abelian surfaces A _{f} attached to cusp forms of Neben type character of level N = 29, 229, 349, 461, and 509.

Original language | English |
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Pages (from-to) | 165-182 |

Number of pages | 18 |

Journal | Manuscripta Mathematica |

Volume | 98 |

Issue number | 2 |

Publication status | Published - 1999 Feb |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

_{2}-type.

*Manuscripta Mathematica*,

*98*(2), 165-182.