Genus 3 curves over ℚ

This page provides download links to lists of 67,879 hyperelliptic curves and 82,240 nonhyperelliptic curves defined over ℚ that have absolute discriminant |D| ≤ 10,000,000. The computation of the hyperelliptic curves was achieved using the methods described in the paper A database of genus 2 curves over the rational numbers which are applicable to hyperelliptic curves of arbitrary genus. The computaiton of the nonhyperelliptic curves is described in the paper A database of nonhyperelliptic genus 3 curves over Q. The download link for hyperelliptic curves provides a colon-delimited text file with the format

D:N:[f(x),h(x)]
where f(x) and h(x) are integer polynomials defining a hyperelliptic curve

y² + h(x)y = f(x)
with absolute descriminant D and conductor N (the conductor of its Jacobian). The download link for nonhyperelliptic curves provides a colon-delimited text file with the format

D:[f(x,y,z)]
where f(x,y,z) is a homogenous quartic polynomial defining a smooth plane curve

f(x,y,z) = 0
with absolute discriminant D (the computations of the conductors of these curves is work in progress). In both cases, each curve is a global minimal model for its isomorphism class.

If you use this data in your research, please cite the relevant papers using these bibliographic details and/or these bibliographic details as appropriate; you can also find this information on my home page.

We remark that not every genus 3 curve over ℚ can be put in of the two forms above. One must also consider degree-two covers of pointless conics, which are geometrically hyperelliptic but do not have a model of the form y² + h(x)y = f(x) that is defined over Q. These curves can instead be described by a pair of equations

g(x,y,z) =0,    w² = f(x,y,z),
in [2,1,1,1]-weighted projective space, where g is a conic and f is a homogenous quartic. A database of genus 3 curves of this form with small discriminants is currently under construction.

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