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: | $10$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8N0 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}39&8\\104&89\end{bmatrix}$, $\begin{bmatrix}59&118\\36&117\end{bmatrix}$, $\begin{bmatrix}67&96\\112&103\end{bmatrix}$, $\begin{bmatrix}105&70\\120&101\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 136.48.0.l.2 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 infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 136.48.0-68.c.1.13 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-68.c.1.16 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-8.e.1.4 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-136.i.2.12 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 136.48.0-136.i.2.20 | $136$ | $2$ | $2$ | $0$ | $?$ |
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.t.2.3 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.y.1.5 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.be.1.5 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.bg.2.2 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.bw.1.6 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.by.2.6 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.cc.2.7 | $136$ | $2$ | $2$ | $1$ |
| 136.192.1-136.cd.1.6 | $136$ | $2$ | $2$ | $1$ |