scholarly journals Index Calculus in Class Groups of Non-hyperelliptic Curves of Genus Three

2007 ◽  
Vol 21 (4) ◽  
pp. 593-611 ◽  
Author(s):  
Claus Diem ◽  
Emmanuel Thomé
2016 ◽  
Vol 19 (A) ◽  
pp. 220-234 ◽  
Author(s):  
David Harvey ◽  
Maike Massierer ◽  
Andrew V. Sutherland

Let$C/\mathbf{Q}$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$\mathbf{Q}$, but may not have a hyperelliptic model of the usual form over$\mathbf{Q}$. 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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