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


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