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: | $\begin{bmatrix}21&48\\112&47\end{bmatrix}$, $\begin{bmatrix}45&116\\90&73\end{bmatrix}$, $\begin{bmatrix}67&132\\12&69\end{bmatrix}$, $\begin{bmatrix}81&104\\56&19\end{bmatrix}$, $\begin{bmatrix}109&12\\74&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 136.24.0.h.2 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $36$ |
| Cyclic 136-torsion field degree: | $2304$ |
| 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.1.9 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 136.24.0-4.b.1.8 | $136$ | $2$ | $2$ | $0$ | $?$ |
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.a.1.13 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.b.1.13 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.d.1.10 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.e.1.10 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.i.2.10 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.k.1.16 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.m.1.10 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.o.2.10 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.q.2.15 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.s.1.11 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.u.2.11 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.w.2.12 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.y.1.10 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.z.2.14 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bb.2.12 | $136$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bc.2.10 | $136$ | $2$ | $2$ | $0$ |
| 136.96.1-136.m.1.2 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.q.1.16 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.w.1.8 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.x.1.6 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bc.2.16 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.be.1.9 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bg.1.6 | $136$ | $2$ | $2$ | $1$ |
| 136.96.1-136.bi.2.8 | $136$ | $2$ | $2$ | $1$ |