Torsion group G               |G|       Parameterizing variety       All hyperelliptic?
=======================================   ===   ==============================   ==================

             Z/2Z x Z/30Z                  60    positive rank elliptic curve           yes

             Z/10Z x Z/10Z                100                P^1                        yes

          Z/2Z x Z/8Z x Z/8Z              128   positive rank elliptic surface          yes

          Z/4Z x Z/4Z x Z/8Z              128                P^1                        yes

             Z/4Z x Z/40Z                 160    positive rank elliptic curve            no

          Z/2Z x Z/4Z x Z/24Z             192    positive rank elliptic curve            no

      Z/2Z x Z/2Z x Z/2Z x Z/24Z          192   positive rank elliptic surface          yes

             Z/10Z x Z/20Z                200                P^2                         no

          Z/6Z x Z/6Z x Z/6Z              216    positive rank elliptic curve            no

             Z/4Z x Z/60Z                 240    positive rank elliptic curve            no

          Z/4Z x Z/8Z x Z/8Z              256    positive rank elliptic curve            no

       Z/2Z x Z/2Z x Z/8Z x Z/8Z          256                P^2                         no

       Z/2Z x Z/4Z x Z/4Z x Z/8Z          256                P^2                         no

   Z/2Z x Z/2Z x Z/2Z x Z/4Z x Z/8Z       256                P^2                        yes

         Z/2Z x Z/12Z x Z/12Z             288                P^2                         no

      Z/2Z x Z/2Z x Z/6Z x Z/12Z          288   positive rank elliptic surface          yes

   Z/2Z x Z/2Z x Z/4Z x Z/4Z x Z/8Z       512    positive rank elliptic curve            no

Z/2Z x Z/2Z x Z/2Z x Z/2Z x Z/4Z x Z/8Z   512                P^1                        yes

         Z/6Z x Z/12Z x Z/12Z             864                P^0                         no

===================================================================================================
               Families of curves over Q of genus 3 such that G is contained in the
              torsion subgroup of the Jacobian. The final column indicates whether or
                    not the family consists entirely of hyperelliptic curves.

Return to the bibliography.