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.
| 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 |
P1 |
yes |
| Z/2Z x Z/8Z x Z/8Z |
128 |
positive rank elliptic surface |
yes |
| Z/4Z x Z/4Z x Z/8Z |
128 |
P1 |
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 |
P2 |
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 |
P2 |
no |
| Z/2Z x Z/4Z x Z/4Z x Z/8Z |
256 |
P2 |
no |
| Z/2Z x Z/2Z x Z/2Z x Z/4Z x Z/8Z |
256 |
P2 |
yes |
| Z/2Z x Z/12Z x Z/12Z |
288 |
P2 |
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 |
P1 |
yes |
| Z/6Z x Z/12Z x Z/12Z |
864 |
P0 |
no |