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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 Q(ζm), even though the equation we give may have coefficients in a smaller field (usually the maximal real subfield of Q(ζm)).
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 Q(ζm)).
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) | g | d |
X1(m,mn) | g | d |
X1(m,mn) | g | d |
X1(m,mn) | g | d |
X1(m,mn) | g | d |
X1(m,mn) | g | d |
X1(m,mn) | g | d |
X1(m,mn) | g | d |
X1(m,mn) | g | d |
| X1(2,2) | 0 | 1 |
X1(3,3) | 0 | 1 |
X1(4,4) | 0 | 1 |
X1(5,5) | 0 | 1 |
X1(6,6) | 1 | 2 |
X1(7,7) | 3 | 5 |
X1(8.8) | 5 | 6 |
X1(9,9) | 10 | 9 |
X1(10,10) | 13 | 10 |
| X1(2,4) | 0 | 1 |
X1(3,6) | 0 | 1 |
X1(4,8) | 1 | 2 |
X1(5,10) | 4 | 6 |
X1(6,12) | 9 | 10 |
X1(7,14) | 19 | 15 |
X1(8.16) | 33 | 24 |
| X1(2,6) | 0 | 1 |
X1(3,9) | 1 | 3 |
X1(4,12) | 5 | 6 |
X1(5,15) | 17 | 16 |
X1(6,18) | 28 | 22 |
| X1(2,8) | 0 | 1 |
X1(3,12) | 3 | 3 |
X1(4,16) | 13 | 8 |
X1(5,20) | 31 | 24 |
| X1(2,10) | 1 | 2 |
X1(3,15) | 9 | 6 |
X1(4,20) | 25 | 12 |
| X1(2,12) | 1 | 2 |
X1(3,18) | 10 | 7 |
X1(4,24) | 33 | 22 |
| X1(2,14) | 4 | 3 |
X1(3,21) | 25 | 12 |
X1(4,28) | 61 | 32 |
| X1(2,16) | 5 | 4 |
X1(3,24) | 25 | 12 |
X1(4,32) | 81 | 32 |
| X1(2,18) | 7 | 6 |
X1(3,27) | 46 | 20 |
X1(4,36) | 97 | 36 |
| X1(2,20) | 9 | 8 |
X1(3,30) | 41 | 18 |
| X1(2,22) | 16 | 9 |
X1(3,33) | 81 | 30 |
| X1(2,24) | 13 | 9 |
X1(3,36) | 64 | 27 |
| X1(2,26) | 25 | 12 |
X1(3,39) | 121 | 42 |
| X1(2,28) | 25 | 15 |
| X1(2,30) | 25 | 15 |
| X1(2,32) | 37 | 19 |
| X1(2,34) | 49 | 20 |
| X1(2,36) | 41 | 20 |
| X1(2,38) | 64 | 26 |
| X1(2,40) | 57 | 29 |
| X1(2,42) | 61 | 30 |
| X1(2,44) | 81 | 35 |
| X1(2,46) | 100 | 39 |
| X1(2,48) | 81 | 39 |
| X1(2,50) | 109 | 45 |
| X1(2,52) | 121 | 51 |
| X1(2,54) | 118 | 48 |
| X1(2,56) | 133 | 57 |
| X1(2,58) | 169 | 63 |
| X1(2,60) | 129 | 57 |
This work was supported by NSF grant DMS-1522526.