Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $1^{2}\cdot4$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4B0 |
| Rouse and Zureick-Brown (RZB) label: | X9a |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.12.0.24 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&7\\4&5\end{bmatrix}$, $\begin{bmatrix}5&2\\2&7\end{bmatrix}$, $\begin{bmatrix}7&0\\6&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3:\OD_{16}$ |
| Contains $-I$: | no $\quad$ (see 4.6.0.b.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $4$ |
| Cyclic 8-torsion field degree: | $16$ |
| Full 8-torsion field degree: | $128$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 11629 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(x^{2}+48y^{2})^{3}}{y^{4}x^{6}(x^{2}+64y^{2})}$ |
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.a.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-4.c.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.c.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.h.1.1 | $8$ | $2$ | $2$ | $0$ |
| 24.24.0-12.e.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-12.f.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.m.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.p.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.36.1-12.b.1.9 | $24$ | $3$ | $3$ | $1$ |
| 24.48.0-12.f.1.5 | $24$ | $4$ | $4$ | $0$ |
| 40.24.0-20.e.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-20.f.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.m.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.p.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.60.2-20.b.1.7 | $40$ | $5$ | $5$ | $2$ |
| 40.72.1-20.b.1.15 | $40$ | $6$ | $6$ | $1$ |
| 40.120.3-20.b.1.6 | $40$ | $10$ | $10$ | $3$ |
| 56.24.0-28.e.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-28.f.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.m.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.p.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.96.2-28.b.1.10 | $56$ | $8$ | $8$ | $2$ |
| 56.252.7-28.b.1.11 | $56$ | $21$ | $21$ | $7$ |
| 56.336.9-28.b.1.2 | $56$ | $28$ | $28$ | $9$ |
| 72.324.10-36.c.1.8 | $72$ | $27$ | $27$ | $10$ |
| 88.24.0-44.e.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-44.f.1.3 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.m.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.p.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.144.4-44.b.1.10 | $88$ | $12$ | $12$ | $4$ |
| 104.24.0-52.e.1.3 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-52.f.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.m.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.p.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.168.5-52.b.1.14 | $104$ | $14$ | $14$ | $5$ |
| 120.24.0-60.e.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-60.f.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.m.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.p.1.8 | $120$ | $2$ | $2$ | $0$ |
| 136.24.0-68.e.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-68.f.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.m.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.p.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.216.7-68.b.1.13 | $136$ | $18$ | $18$ | $7$ |
| 152.24.0-76.e.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-76.f.1.3 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.m.1.4 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.p.1.4 | $152$ | $2$ | $2$ | $0$ |
| 152.240.8-76.b.1.11 | $152$ | $20$ | $20$ | $8$ |
| 168.24.0-84.e.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-84.f.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.m.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.p.1.4 | $168$ | $2$ | $2$ | $0$ |
| 184.24.0-92.e.1.4 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-92.f.1.4 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.m.1.4 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.p.1.4 | $184$ | $2$ | $2$ | $0$ |
| 184.288.10-92.b.1.12 | $184$ | $24$ | $24$ | $10$ |
| 232.24.0-116.e.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-116.f.1.1 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-232.m.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.24.0-232.p.1.4 | $232$ | $2$ | $2$ | $0$ |
| 232.360.13-116.b.1.15 | $232$ | $30$ | $30$ | $13$ |
| 248.24.0-124.e.1.4 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-124.f.1.3 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-248.m.1.4 | $248$ | $2$ | $2$ | $0$ |
| 248.24.0-248.p.1.4 | $248$ | $2$ | $2$ | $0$ |
| 248.384.14-124.b.1.11 | $248$ | $32$ | $32$ | $14$ |
| 264.24.0-132.e.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-132.f.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.m.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.24.0-264.p.1.4 | $264$ | $2$ | $2$ | $0$ |
| 280.24.0-140.e.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-140.f.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.m.1.8 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0-280.p.1.8 | $280$ | $2$ | $2$ | $0$ |
| 296.24.0-148.e.1.1 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-148.f.1.3 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-296.m.1.4 | $296$ | $2$ | $2$ | $0$ |
| 296.24.0-296.p.1.4 | $296$ | $2$ | $2$ | $0$ |
| 296.456.17-148.b.1.9 | $296$ | $38$ | $38$ | $17$ |
| 312.24.0-156.e.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-156.f.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.m.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.24.0-312.p.1.8 | $312$ | $2$ | $2$ | $0$ |
| 328.24.0-164.e.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-164.f.1.3 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-328.m.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.24.0-328.p.1.1 | $328$ | $2$ | $2$ | $0$ |
| 328.504.19-164.b.1.12 | $328$ | $42$ | $42$ | $19$ |