Families of curves over Q of genus 2 such that G is contained in the torsion subgroup of the Jacobian.
Torsion group G |G| Parameterizing variety
Z/20Z 20 P2
Z/21Z 21 P2
Z/3Z x Z/9Z 27 P2
Z/30Z 30 P2
Z/35Z 35 positive rank elliptic curve
Z/6Z x Z/6Z 36 P2
Z/3Z x Z/12Z 36 P2
Z/40Z 40 positive rank elliptic surface
Z/45Z 45 positive rank elliptic curve
Z/2Z x Z/24Z 48 P2
Z/7Z x Z/7Z 49 P0
Z/5Z x Z/10Z 50 positive rank elliptic surface
Z/60Z 60 positive rank elliptic curve
Z/63Z 63 P0
Z/8Z x Z/8Z 64 P2
Z/2Z x Z/4Z x Z/8Z 64 P2
Z/6Z x Z/12Z 72 positive rank elliptic surface
Z/2Z x Z/6Z x Z/6Z 72 positive rank elliptic surface
Z/2Z x Z/2Z x Z/24Z 96 positive rank elliptic curve
Z/2Z x Z/2Z x Z/4Z x Z/8Z 128 positive rank elliptic surface


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