Abstract
We construct a parametric family {E(±)(s,t,u)} of minimal Q-curves of degree 5 over the quadratic fields Q\(\), and the family {C(s,t,u)} of genus two curves over Q covering E{(+)(s,t,u) whose jacobians are abelian surfaces of GL2-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 Af attached to cusp forms of Neben type character of level N= 29, 229, 349, 461, and 509.
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