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: | $\begin{bmatrix}81&120\\122&53\end{bmatrix}$, $\begin{bmatrix}85&32\\36&7\end{bmatrix}$, $\begin{bmatrix}97&96\\94&31\end{bmatrix}$, $\begin{bmatrix}129&0\\128&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 136.48.0.bd.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $18$ |
| Cyclic 136-torsion field degree: | $1152$ |
| 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.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 136.48.0-8.i.1.11 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-136.i.1.3 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-136.i.1.13 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-136.cb.2.4 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-136.cb.2.13 | $136$ | $2$ | $2$ | $0$ | $?$ |
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.q.1.2 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.br.1.1 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.cc.1.2 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.cg.1.1 | $136$ | $2$ | $2$ | $1$ |
| 272.192.1-272.c.1.4 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.l.1.2 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.o.1.3 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.r.1.3 | $272$ | $2$ | $2$ | $1$ |
| 272.192.2-272.o.1.3 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-272.p.1.1 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-272.q.1.1 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-272.r.1.3 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-272.s.1.1 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-272.t.1.3 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-272.u.1.3 | $272$ | $2$ | $2$ | $2$ |
| 272.192.2-272.v.1.1 | $272$ | $2$ | $2$ | $2$ |
| 272.192.3-272.cg.2.1 | $272$ | $2$ | $2$ | $3$ |
| 272.192.3-272.cl.2.3 | $272$ | $2$ | $2$ | $3$ |
| 272.192.3-272.dg.2.1 | $272$ | $2$ | $2$ | $3$ |
| 272.192.3-272.dq.2.1 | $272$ | $2$ | $2$ | $3$ |