README FILE FOR GENUS 2 CURVES DATA FILES
=========================================

The files in this directory contain data on genus 2 curves with
small odd discriminant and small coefficients. All these curves
have good reduction mod 2, and one can show that every curve with
good reduction mod 2 has a model
  y^2 + h(x) y = f_0(x) + 2 f(x) ,
where (h(x), f_0(x)) is some pair in a list of 160 pairs of polynomials
with 0-1-coefficients of degrees (<=3, <=6) and f(x) is some 
polynomial with integral coefficients. The Magma code for producing
the lists is given below. 

curves-with-small-odd-discriminant.txt
======================================

This file contains all curves as above with f with coefficients
bounded by 5 in absolute value and discriminant at most 11^4
(obtained by calling FindCurves(5, 11^4)). The curves are sorted
(1) by absolute value of the discriminant
(2) by their Euler factor at 2
(3) by isomorphism classes.

As an example, consider the entry for discriminant 5497:

//======================================================================
// disc = 5497                                      <-- discriminant
//======================================================================
// Euler factor at 2 = 4*T^4 + 2*T^3 + T^2 + T + 1  <-- Euler factor
//----------------------------------------------------------------------
// Isomorphism Class 1                              <-- number of iso. class
// y^2 + (x^3 + x + 1)*y = x^3                      <-- some small model
// y^2 = x^6 + 2*x^4 + 6*x^3 + x^2 + 2*x + 1        <-- a model y^2 = f(x)
//----------------------------------------------------------------------
  <x^3 + x + 1, -2*x^3 - x, 5497>,                  <-- all models <h, f, d>
  <x^3 + x + 1, x^3, 5497>,                         <-- ...
// Isomorphism Class 2
// y^2 + (x^3 + x + 1)*y = -x^5 + x^4 - 5*x^2 + 6*x - 4
// y^2 = x^6 - 4*x^5 + 6*x^4 + 2*x^3 - 19*x^2 + 26*x - 15
//----------------------------------------------------------------------
  <x^3 + x + 1, -x^5 + x^4 - 5*x^2 + 6*x - 4, 5497>,
  <x^3 + x + 1, x^5 + x^4 - x^3 - 5*x^2 - 7*x - 4, 5497>,
//----------------------------------------------------------------------
// Euler factor at 2 = 4*T^4 - 2*T^3 + T^2 - T + 1
//----------------------------------------------------------------------
// Isomorphism Class 1
// y^2 + (x^3 + x + 1)*y = -x^6 + x^5 + 2*x^4 + 3*x^3 + 3*x^2 + x
// y^2 = -3*x^6 + 4*x^5 + 10*x^4 + 14*x^3 + 13*x^2 + 6*x + 1
//----------------------------------------------------------------------
  <x^3 + x + 1, -x^6 + x^5 + 2*x^4 + 3*x^3 + 3*x^2 + x, 5497>,
  <x^3 + x + 1, -x^6 + x^5 - x^4 - 3*x^3 + 3*x^2 - 2*x, 5497>,
  <x^3 + x + 1, -x^6 - x^5 - x^4 + 2*x^3 + 3*x^2 + x, 5497>,
  <x^3 + x + 1, -x^6 - x^5 + 2*x^4 - 4*x^3 + 3*x^2 - 2*x, 5497>,
//----------------------------------------------------------------------
// Euler factor at 2 = 4*T^4 - T^2 + 1
//----------------------------------------------------------------------
// Isomorphism Class 1
// y^2 + (x^2 + x)*y = -x^5 - x^4 + 2*x^3 - x
// y^2 = -4*x^5 - 3*x^4 + 10*x^3 + x^2 - 4*x
//----------------------------------------------------------------------
  <x^2 + x, -x^5 - 5*x^4 - 7*x^3 - 2*x^2 + x, 5497>,
  <x^2 + x, x^5 - x^4 - 3*x^3 + x, 5497>,
  <x^2 + x, x^5 - 7*x^4 + 10*x^3 - 6*x^2 + x, 5497>,
  <x^2 + x, x^5 - 5*x^4 + 6*x^3 - 2*x^2 - x, 5497>,
  <x^2 + x, -x^6 + 4*x^5 - 2*x^4 - 5*x^3 + 4*x^2 - x, 5497>,
  <x^2 + x, -x^5 + 5*x^4 - 9*x^3 + 3*x^2 + 2*x - 1, 5497>,
  <x^2 + x, -x^5 - x^4 + 2*x^3 - x, 5497>,
  <x^2 + x, -x^6 - 4*x^5 - 2*x^4 + 4*x^3 + 4*x^2 + x, 5497>,
  <x^2 + x, x^5 + 5*x^4 + 8*x^3 + 3*x^2 - 2*x - 1, 5497>,


small-curves-with-more-info.txt
===============================

This file contains the first bunch of curves (up to discriminant 4000),
together with additional arithmetic information -- Mordell-Weil rank,
torsion subgroup, number of rational points, plus conductor and
reduction types (up to disc. 1205; thanks to Qing Liu's genus2reduction 
program).

Magma code for the search
=========================

The code is simple and straightforward (maybe apart from the list
of (h, f_0) pairs).

------------------------------------------------------------------------
P<x> := PolynomialRing(Rationals());

legalQ := 
[
    <P!1, [
        x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,
        x^5 + x^3 + x^2,
        x^5 + x^2,
        x^5 + x^4 + x^3 + x^2,
        x^5 + x^2 + 1,
        x^5 + x^3 + x^2 + x,
        x^6 + x^5 + 1,
        x^6 + x^5 + x^4 + x^3,
        x^5 + x^4 + x^2,
        x^6 + x^5 + x^4 + x + 1,
        x^5 + x^2 + x,
        x^6 + x^5 + x^3,
        x^5 + x^4 + x^3 + x^2 + 1,
        x^5 + x^3 + x^2 + x + 1,
        x^6 + x^5 + x^4 + x^3 + x,
        x^6 + x^5 + x^4 + x^2 + 1,
        x^5 + x^4 + x^3 + x^2 + x,
        x^6 + x^5 + x^3 + x,
        x^6 + x^5 + x^4 + x^3 + x^2,
        x^5 + x^4 + x^2 + x,
        x^6 + x^5 + x^3 + x^2,
        x^6 + x^5 + x^2 + x + 1,
        x^5 + x^3,
        x^6 + x^5 + x^4 + x^3 + x^2 + x,
        x^5 + x^4 + x^2 + x + 1,
        x^6 + x^5 + x^3 + x^2 + x,
        x^5,
        x^5 + x^3 + 1,
        x^5 + x^4 + x^3,
        x^5 + x^3 + x,
        x^6 + x^5 + x^4,
        x^5 + x^4,
        x^5 + x,
        x^6 + x^5,
        x^6 + x^5 + x^4 + x,
        x^5 + x^4 + 1,
        x^5 + x + 1,
        x^5 + x^4 + x^3 + x,
        x^6 + x^5 + x,
        x^6 + x^5 + x^4 + x^2,
        x^5 + x^4 + x,
        x^5 + x^4 + x^3 + x + 1,
        x^6 + x^5 + x^2,
        x^6 + x^5 + x^4 + x^2 + x,
        x^6 + x^5 + x^4 + x^3 + 1,
        x^6 + x^5 + x^2 + x,
        x^6 + x^5 + x^3 + x + 1,
        x^6 + x^5 + x^3 + x^2 + 1
    ]>,
    <x, [
        x^5 + x^4 + 1,
        x^5 + x,
        x^6 + x^5 + x^4 + x^3 + x^2 + x,
        x^6 + x^5 + x^4 + x^3 + 1,
        x^6 + x^5 + x^4 + x^2 + x,
        x^5 + x^4 + x^2 + x,
        x^6 + x^5 + x^4 + 1,
        x^5 + x^4 + x,
        x^5 + x^3 + x^2 + 1,
        x^6 + x^5 + x^3 + x,
        x^5 + x^3 + 1,
        x^6 + x^5 + x^4 + x^3 + x^2 + 1,
        x^6 + x^5 + x,
        x^5 + x^4 + x^3 + x^2 + 1,
        x^5 + x^3 + x^2 + x,
        x^6 + x^5 + x^4 + x^2 + 1,
        x^5 + x^4 + x^3 + 1,
        x^5 + x^3 + x,
        x^6 + x^5 + x^3 + x^2 + x,
        x^6 + x^5 + x^3 + 1,
        x^5 + x^4 + x^3 + x^2 + x,
        x^6 + x^5 + x^2 + x,
        x^6 + x^5 + 1,
        x^5 + x^4 + x^3 + x,
        x^5 + x^2 + 1,
        x^6 + x^5 + x^4 + x^3 + x,
        x^5 + 1,
        x^6 + x^5 + x^4 + x,
        x^6 + x^5 + x^3 + x^2 + 1,
        x^5 + x^4 + x^2 + 1,
        x^5 + x^2 + x,
        x^6 + x^5 + x^2 + 1
    ]>,
    <x^2 + x, [
        x^6 + x^2 + 1,
        x^6 + x,
        x^5 + x^4 + x^3 + x,
        x^5 + x^4 + x^3 + x^2 + 1,
        x^6 + x^3 + x^2 + 1,
        x^5 + x^4 + x,
        x^5 + x^4 + x^2 + 1,
        x^6 + x^3 + x
    ]>,
    <x^3 + 1, [
        x^6 + x^5 + x^4 + x^3 + x^2,
        x^6 + x^5 + x^2 + x + 1,
        x^6 + x^4 + x + 1,
        x^6 + x^3,
        x^6 + x^5 + x^4 + x^2,
        x^6 + x^5 + x^4 + x^3 + 1,
        x^6,
        x^3 + x^2 + x,
        x^5 + x^3 + x^2,
        x^6 + x^5 + x^4 + 1,
        x^5 + x^4 + x,
        x^6 + x^5 + x^3 + x,
        x^6 + x^5 + x,
        x^6 + x^3 + x^2 + 1,
        x^5 + x^2,
        x^5 + x^4 + x^3 + x,
        x^4,
        x^2 + x,
        x^6 + x^2 + 1,
        x^6 + x^4 + x^3 + x^2 + x,
        x^6 + x^5 + x^3 + x^2 + x + 1,
        x^4 + x^3,
        x^6 + x^4 + x^2 + x,
        x^6 + x^4 + x^3 + x + 1
    ]>,
    <x^3 + x + 1, [
        x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,
        x^6 + x^4 + 1,
        x^5 + x^3 + x^2,
        x^6 + x^5 + x^3 + x^2 + x + 1,
        x^6 + x^4 + x^3,
        x^2 + x,
        x^6 + 1,
        x^6 + x^4 + x + 1,
        x^4 + x^2,
        x^6 + x + 1,
        x^6 + x^3 + x,
        x^4,
        x^5 + x^4 + x^2,
        x^5 + x^2 + x,
        x^4 + x^3 + x^2,
        x^6 + x^5 + x^4 + x^2 + 1,
        x^5 + x^4 + x^3 + x^2 + x,
        x^6 + x^5 + x^4 + x^3 + x^2,
        x^6 + x^5 + x^2 + 1,
        x^6 + x^5 + x^4 + x^2 + x + 1,
        x^6 + x^5 + x^2 + x + 1,
        x^5 + x^3,
        x^6 + x^5 + x^3 + x^2 + x,
        x^5,
        x^6,
        x,
        x^6 + x^4 + x,
        x^5 + x^4 + x^3,
        x^3,
        x^5 + x^3 + x,
        x^5 + x^4,
        x^5 + x,
        x^3 + x^2 + x,
        x^3 + x^2,
        x^5 + x^4 + x^3 + x,
        x^6 + x^4 + x^3 + 1,
        x^5 + x^4 + x,
        x^6 + x^3 + 1,
        x^6 + x^5 + x^2,
        x^6 + x^4 + x^3 + x + 1,
        x^6 + x^5 + x^4 + x^2 + x,
        x^4 + x^2 + x,
        x^6 + x^3 + x + 1,
        x^4 + x^3 + x^2 + x,
        x^6 + x^5 + x^4 + x^3 + x^2 + 1,
        x^2,
        x^4 + x^3 + x,
        x^6 + x^5 + x^3 + x^2 + 1
    ]>
];

procedure FindCurves(n, b)
// n: add -2n, -(2n-2), ..., 0, 2, ..., 2n to the coefficients of f0
// b: upper bound for absolute value of discriminant
  s := {-2*n..2*n by 2};
  for l in legalQ do
    h := l[1]; h2 := h*h;
    for f0 in l[2] do
      for cs0,cs1,cs2,cs3,cs4,cs5,cs6 in s do
        cs := [cs0,cs1,cs2,cs3,cs4,cs5,cs6];
        f := f0 + P!cs;
        k := h2 + 4*f;
        d := Discriminant(k);
        if Degree(k) lt 6 then d *:= LeadingCoefficient(k)^2; end if;
        d /:= 2^12;
        if b eq 0 or Abs(d) le b then
          printf "<%o, %o, %o>\n", h, f, d;
        end if;
      end for;
    end for;
  end for;
end procedure;
-----------------------------------------------------------------------

Michael Stoll <m.stoll@iu-bremen.de>