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: | $6$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 8J0 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}63&84\\36&7\end{bmatrix}$, $\begin{bmatrix}69&92\\86&19\end{bmatrix}$, $\begin{bmatrix}69&124\\24&115\end{bmatrix}$, $\begin{bmatrix}107&112\\52&97\end{bmatrix}$, $\begin{bmatrix}115&84\\40&107\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 136.24.0.i.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $36$ |
| Cyclic 136-torsion field degree: | $1152$ |
| Full 136-torsion field degree: | $2506752$ |
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 |
|---|---|---|---|---|---|
| 8.24.0-4.b.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 136.24.0-4.b.1.3 | $136$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 136.96.0-136.b.1.23 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.c.1.8 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.e.2.11 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.f.1.14 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.j.2.11 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.l.1.12 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.n.1.12 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.p.2.10 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.r.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.t.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.v.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.x.2.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.z.1.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.ba.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bc.2.3 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bd.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.96.1-136.q.1.7 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.s.2.9 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.x.2.7 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.y.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bd.2.5 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bf.1.4 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bh.1.3 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bj.2.7 | $136$ | $2$ | $2$ | $1$ |