Invariants
| Level: | $136$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}17&80\\74&121\end{bmatrix}$, $\begin{bmatrix}105&88\\95&91\end{bmatrix}$, $\begin{bmatrix}123&68\\14&27\end{bmatrix}$, $\begin{bmatrix}123&88\\135&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 136.24.0.bf.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $36$ |
| Cyclic 136-torsion field degree: | $2304$ |
| Full 136-torsion field degree: | $2506752$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.d.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 68.24.0-4.d.1.1 | $68$ | $2$ | $2$ | $0$ | $0$ |
| 136.24.0-136.bb.1.3 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.24.0-136.bb.1.7 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.24.0-136.bb.1.10 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.24.0-136.bb.1.14 | $136$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 136.96.1-136.ea.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.eb.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.ec.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.ed.1.4 | $136$ | $2$ | $2$ | $1$ |
| 272.96.0-272.t.1.2 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.t.1.8 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.u.1.2 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.u.1.8 | $272$ | $2$ | $2$ | $0$ |
| 272.96.2-272.p.1.2 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2-272.p.1.16 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2-272.s.1.3 | $272$ | $2$ | $2$ | $2$ |
| 272.96.2-272.s.1.16 | $272$ | $2$ | $2$ | $2$ |