Invariants
| Level: | $136$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-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 | $2\cdot4$ | ||
| 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: | $\begin{bmatrix}57&44\\82&125\end{bmatrix}$, $\begin{bmatrix}107&72\\129&97\end{bmatrix}$, $\begin{bmatrix}111&86\\132&109\end{bmatrix}$, $\begin{bmatrix}125&6\\11&131\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 272.48.0-136.cc.1.1, 272.48.0-136.cc.1.2, 272.48.0-136.cc.1.3, 272.48.0-136.cc.1.4, 272.48.0-136.cc.1.5, 272.48.0-136.cc.1.6, 272.48.0-136.cc.1.7, 272.48.0-136.cc.1.8 |
| Cyclic 136-isogeny field degree: | $72$ |
| Cyclic 136-torsion field degree: | $4608$ |
| Full 136-torsion field degree: | $5013504$ |
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 |
|---|---|---|---|---|---|
| 8.12.0.q.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 68.12.0.k.1 | $68$ | $2$ | $2$ | $0$ | $0$ |
| 136.12.0.bt.1 | $136$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 136.48.1.eu.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.ev.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.fc.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.fd.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.hi.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.hj.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.hq.1 | $136$ | $2$ | $2$ | $1$ |
| 136.48.1.hr.1 | $136$ | $2$ | $2$ | $1$ |