Optimized equations for X1(m,mn)


The table below gives links to optimized equations f(u,v)=0 for X1(m,mn), together with parametizations
E=[a1(u,v),a2(u,v),a3(u,v),a4(u,v),a6(u,v)],     P=[Px(u,v),Py(u,v)],     Q=[Qx(u,v),Qy(u,v)],
that define an elliptic curve
y2 + a1xy + a3y = x3 + a2x2 + a4x + a6
in Weierstrass form on which P is a point of order m and Q is a point of order mn. Note that the curve X1(m,mn) is defined over the cyclotomic field Qm), even though the equation we give may have coefficients in a smaller field (usually the maximal real subfield of Qm)). The value g listed in the table is the genus X1(m,mn), and the value d is the minimal (nonzero) degree of u or v in the given equation (this gives an upper bound on the gonality of the curve over Qm)).

The method used to construct these models is described in the paper Torsion subgroups of elliptic curves over quintic and sextic number fields, with Maarten Derickx, Proceedings of the AMS, 145 (2017), 4233-4245. Please be sure to cite this paper if you use these models in your research.

For m=1, see the equations for X1(n).

X1(m,mn)gd X1(m,mn)gd X1(m,mn)gd X1(m,mn)gd X1(m,mn)gd X1(m,mn)gd X1(m,mn)gd X1(m,mn)gd X1(m,mn)gd
X1(2,2)01 X1(3,3)01 X1(4,4)01 X1(5,5)01 X1(6,6)12 X1(7,7)35 X1(8.8)56 X1(9,9)109 X1(10,10)1310
X1(2,4)01 X1(3,6)01 X1(4,8)12 X1(5,10)46 X1(6,12)910 X1(7,14)1915 X1(8.16)3324
X1(2,6)01 X1(3,9)13 X1(4,12)56 X1(5,15)1716 X1(6,18)2822
X1(2,8)01 X1(3,12)33 X1(4,16)138 X1(5,20)3124
X1(2,10)12 X1(3,15)96 X1(4,20)2512
X1(2,12)12 X1(3,18)107 X1(4,24)3322
X1(2,14)43 X1(3,21)2512 X1(4,28)6132
X1(2,16)54 X1(3,24)2512 X1(4,32)8132
X1(2,18)76 X1(3,27)4620 X1(4,36)9736
X1(2,20)98 X1(3,30)4118
X1(2,22)169 X1(3,33)8130
X1(2,24)139 X1(3,36)6427
X1(2,26)2512 X1(3,39)12142
X1(2,28)2515
X1(2,30)2515
X1(2,32)3719
X1(2,34)4920
X1(2,36)4120
X1(2,38)6426
X1(2,40)5729
X1(2,42)6130
X1(2,44)8135
X1(2,46)10039
X1(2,48)8139
X1(2,50)10945
X1(2,52)12151
X1(2,54)11848
X1(2,56)13357
X1(2,58)16963
X1(2,60)12957


This work was supported by NSF grant DMS-1522526.