# Computing *L*-series of geometrically hyperelliptic curves of genus three

(with Maike Massierer and Andrew Sutherland)

LMS J. Comput. Math. **19** (2016), suppl. A, 220–234 (DOI)

arXiv preprint (May 2016).

## Abstract

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.

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