Invariants
| Level: | $136$ | $\SL_2$-level: | $8$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $8^{12}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B3 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}17&60\\84&121\end{bmatrix}$, $\begin{bmatrix}57&32\\80&107\end{bmatrix}$, $\begin{bmatrix}65&44\\120&31\end{bmatrix}$, $\begin{bmatrix}113&44\\80&75\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 136.96.3.be.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $36$ |
| Cyclic 136-torsion field degree: | $576$ |
| Full 136-torsion field degree: | $626688$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.96.0-8.c.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 136.96.0-8.c.1.4 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.96.1-136.n.1.2 | $136$ | $2$ | $2$ | $1$ | $?$ |
| 136.96.1-136.n.1.8 | $136$ | $2$ | $2$ | $1$ | $?$ |
| 136.96.2-136.a.1.16 | $136$ | $2$ | $2$ | $2$ | $?$ |
| 136.96.2-136.a.1.22 | $136$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 136.384.5-136.z.1.2 | $136$ | $2$ | $2$ | $5$ |
| 136.384.5-136.z.2.3 | $136$ | $2$ | $2$ | $5$ |
| 136.384.5-136.bb.3.1 | $136$ | $2$ | $2$ | $5$ |
| 136.384.5-136.bb.4.1 | $136$ | $2$ | $2$ | $5$ |
| 272.384.7-272.c.1.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.f.1.2 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.bd.1.10 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.be.1.2 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.cn.1.2 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.co.1.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.do.1.3 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.dr.1.1 | $272$ | $2$ | $2$ | $7$ |