The Torsion Subgroup of Elliptic Curves

Mazur proved that given an elliptic curve defined over the rationals, the only possible torsion subgroups are the following:

Z/NZ with 1<N<11 or N=12

Z/2Z + Z/2NZ with 0<N<5

See this for more info .

Here we show examples of curves with the torsion subgroups mentioned above:

CURVE	TORSION SUBGROUP	GENERATORS
y^2=x^3-2	trivial	O
y^2=x^3+8	Z/2Z	[[-2,0]]
y^2=x^3+4	Z/3Z	[[0,2]]
y^2=x^3+4x	Z/4Z	[[2,4]]
y^2-y=x^3-x^2	Z/5Z	[[0,1]]
y^2=x^3+1	Z/6Z	[[2,3]]
y^2=x^3-43x+166	Z/7Z	[[3,8]]
y^2+7xy=x^3+16x	Z/8Z	[[-2,10]]
y^2+xy+y=x^3-x^2-14x+29	Z/9Z	[[3,1]]
y^2+xy=x^3-45x+81	Z/10Z	[[0,9]]
y^2+43xy-210y=x^3-210x^2	Z/12Z	[[0,210]]
y^2=x^3-4x	Z/2Z + Z/2Z	[[2, 0], [0, 0]]
y^2=x^3+2x^2-3x	Z/4Z + Z/2Z	[[3,6],[0,0]]
y^2+5xy-6y=x^3-3x^2	Z/6Z + Z/2Z	[[-3, 18], [2, -2]]
y^2 +17xy -120y=x^3 -60x^2	Z/8Z + Z/2Z	[[30, -90], [-40, 400]]

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