Properties

Label 8.12.0-4.a.1.1
Level 88
Index 1212
Genus 00
Analytic rank 00
Cusps 33
Q\Q-cusps 11

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Invariants

Level: 88 SL2\SL_2-level: 44
Index: 1212 PSL2\PSL_2-index:66
Genus: 0=1+6120403320 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}
Cusps: 33 (of which 11 is rational) Cusp widths 232^{3} Cusp orbits 121\cdot2
Elliptic points: 00 of order 22 and 00 of order 33
Q\Q-gonality: 11
Q\overline{\Q}-gonality: 11
Rational cusps: 11
Rational CM points: yes (D=\quad(D = 4-4)

Other labels

Cummins and Pauli (CP) label: 2C0
Rouse and Zureick-Brown (RZB) label: X10a
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.12.0.37

Level structure

GL2(Z/8Z)\GL_2(\Z/8\Z)-generators: [3545]\begin{bmatrix}3&5\\4&5\end{bmatrix}, [5521]\begin{bmatrix}5&5\\2&1\end{bmatrix}, [7443]\begin{bmatrix}7&4\\4&3\end{bmatrix}
GL2(Z/8Z)\GL_2(\Z/8\Z)-subgroup: C24.D4C_2^4.D_4
Contains I-I: no \quad (see 4.6.0.a.1 for the level structure with I-I)
Cyclic 8-isogeny field degree: 44
Cyclic 8-torsion field degree: 1616
Full 8-torsion field degree: 128128

Models

This modular curve is isomorphic to P1\mathbb{P}^1.

Rational points

This modular curve has infinitely many rational points, including 11629 stored non-cuspidal points.

Maps to other modular curves

jj-invariant map of degree 6 to the modular curve X(1)X(1) :

j\displaystyle j == 27x6(x24xy+y2)3x6(xy)2(x2+y2)2\displaystyle 2^7\,\frac{x^6 (x^2-4 x y+y^2)^3}{x^6 (x-y)^2 (x^2+y^2)^2}

Modular covers

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
8.24.0-4.c.1.3 88 22 22 00
8.24.0-4.d.1.4 88 22 22 00
8.24.0-8.g.1.2 88 22 22 00
8.24.0-8.j.1.2 88 22 22 00
24.24.0-12.c.1.1 2424 22 22 00
24.24.0-12.d.1.1 2424 22 22 00
24.24.0-24.g.1.1 2424 22 22 00
24.24.0-24.j.1.2 2424 22 22 00
24.36.1-12.a.1.1 2424 33 33 11
24.48.0-12.d.1.7 2424 44 44 00
40.24.0-20.c.1.2 4040 22 22 00
40.24.0-20.d.1.6 4040 22 22 00
40.24.0-40.g.1.1 4040 22 22 00
40.24.0-40.j.1.1 4040 22 22 00
40.60.2-20.a.1.2 4040 55 55 22
40.72.1-20.a.1.8 4040 66 66 11
40.120.3-20.a.1.2 4040 1010 1010 33
56.24.0-28.c.1.1 5656 22 22 00
56.24.0-28.d.1.2 5656 22 22 00
56.24.0-56.g.1.4 5656 22 22 00
56.24.0-56.j.1.1 5656 22 22 00
56.96.2-28.a.1.5 5656 88 88 22
56.252.7-28.a.1.4 5656 2121 2121 77
56.336.9-28.a.1.2 5656 2828 2828 99
72.324.10-36.a.1.2 7272 2727 2727 1010
88.24.0-44.c.1.2 8888 22 22 00
88.24.0-44.d.1.1 8888 22 22 00
88.24.0-88.g.1.1 8888 22 22 00
88.24.0-88.j.1.2 8888 22 22 00
88.144.4-44.a.1.1 8888 1212 1212 44
104.24.0-52.c.1.2 104104 22 22 00
104.24.0-52.d.1.6 104104 22 22 00
104.24.0-104.g.1.1 104104 22 22 00
104.24.0-104.j.1.2 104104 22 22 00
104.168.5-52.a.1.1 104104 1414 1414 55
120.24.0-60.c.1.2 120120 22 22 00
120.24.0-60.d.1.1 120120 22 22 00
120.24.0-120.g.1.2 120120 22 22 00
120.24.0-120.j.1.3 120120 22 22 00
136.24.0-68.c.1.1 136136 22 22 00
136.24.0-68.d.1.1 136136 22 22 00
136.24.0-136.g.1.2 136136 22 22 00
136.24.0-136.j.1.2 136136 22 22 00
136.216.7-68.a.1.7 136136 1818 1818 77
152.24.0-76.c.1.1 152152 22 22 00
152.24.0-76.d.1.1 152152 22 22 00
152.24.0-152.g.1.3 152152 22 22 00
152.24.0-152.j.1.2 152152 22 22 00
152.240.8-76.a.1.7 152152 2020 2020 88
168.24.0-84.c.1.4 168168 22 22 00
168.24.0-84.d.1.4 168168 22 22 00
168.24.0-168.g.1.1 168168 22 22 00
168.24.0-168.j.1.1 168168 22 22 00
184.24.0-92.c.1.2 184184 22 22 00
184.24.0-92.d.1.1 184184 22 22 00
184.24.0-184.g.1.4 184184 22 22 00
184.24.0-184.j.1.1 184184 22 22 00
184.288.10-92.a.1.1 184184 2424 2424 1010
232.24.0-116.c.1.1 232232 22 22 00
232.24.0-116.d.1.1 232232 22 22 00
232.24.0-232.g.1.1 232232 22 22 00
232.24.0-232.j.1.2 232232 22 22 00
232.360.13-116.a.1.8 232232 3030 3030 1313
248.24.0-124.c.1.1 248248 22 22 00
248.24.0-124.d.1.1 248248 22 22 00
248.24.0-248.g.1.2 248248 22 22 00
248.24.0-248.j.1.1 248248 22 22 00
248.384.14-124.a.1.1 248248 3232 3232 1414
264.24.0-132.c.1.1 264264 22 22 00
264.24.0-132.d.1.1 264264 22 22 00
264.24.0-264.g.1.2 264264 22 22 00
264.24.0-264.j.1.1 264264 22 22 00
280.24.0-140.c.1.3 280280 22 22 00
280.24.0-140.d.1.4 280280 22 22 00
280.24.0-280.g.1.2 280280 22 22 00
280.24.0-280.j.1.2 280280 22 22 00
296.24.0-148.c.1.2 296296 22 22 00
296.24.0-148.d.1.4 296296 22 22 00
296.24.0-296.g.1.2 296296 22 22 00
296.24.0-296.j.1.2 296296 22 22 00
296.456.17-148.a.1.3 296296 3838 3838 1717
312.24.0-156.c.1.4 312312 22 22 00
312.24.0-156.d.1.4 312312 22 22 00
312.24.0-312.g.1.2 312312 22 22 00
312.24.0-312.j.1.3 312312 22 22 00
328.24.0-164.c.1.2 328328 22 22 00
328.24.0-164.d.1.4 328328 22 22 00
328.24.0-328.g.1.2 328328 22 22 00
328.24.0-328.j.1.1 328328 22 22 00
328.504.19-164.a.1.3 328328 4242 4242 1919