Invariants
| Level: | $136$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| 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: | 8O0 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}15&40\\22&115\end{bmatrix}$, $\begin{bmatrix}17&128\\14&103\end{bmatrix}$, $\begin{bmatrix}49&100\\14&29\end{bmatrix}$, $\begin{bmatrix}73&124\\34&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 136.48.0.s.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $36$ |
| Cyclic 136-torsion field degree: | $2304$ |
| Full 136-torsion field degree: | $1253376$ |
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 |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.15 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 136.48.0-8.e.1.14 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-136.h.2.23 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-136.h.2.28 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-136.l.1.16 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-136.l.1.17 | $136$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 136.192.1-136.g.2.1 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.j.2.5 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.w.1.2 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.z.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.bc.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.bd.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.bg.2.5 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.bh.2.7 | $136$ | $2$ | $2$ | $1$ |