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.
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