**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

*y*^{2} + *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 *y*^{2} + *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, *w*^{2} = *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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