(with Maike Massierer and Andrew Sutherland)
LMS J. Comput. Math. 19 (2016), suppl. A, 220–234 (DOI)
arXiv preprint (May 2016).
Let C/Q be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of Q, but may not have a hyperelliptic model of the usual form over Q. We describe an algorithm that computes the local zeta functions of C at all odd primes of good reduction up to a prescribed bound N. The algorithm relies on an adaptation of the “accumulating remainder tree” to matrices with entries in a quadratic field. We report on an implementation, and compare its performance to previous algorithms for the ordinary hyperelliptic case.