Invariants
| Level: | $136$ | $\SL_2$-level: | $136$ | Newform level: | $272$ | ||
| Index: | $432$ | $\PSL_2$-index: | $216$ | ||||
| Genus: | $15 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot34^{2}\cdot68^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 15$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 15$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 68D15 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}9&0\\126&33\end{bmatrix}$, $\begin{bmatrix}31&0\\111&61\end{bmatrix}$, $\begin{bmatrix}33&0\\28&57\end{bmatrix}$, $\begin{bmatrix}43&68\\47&73\end{bmatrix}$, $\begin{bmatrix}57&68\\3&63\end{bmatrix}$, $\begin{bmatrix}127&0\\84&123\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 68.216.15.l.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $2$ |
| Cyclic 136-torsion field degree: | $128$ |
| Full 136-torsion field degree: | $278528$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.d.1.2 | $8$ | $18$ | $18$ | $0$ | $0$ |
| $X_0(17)$ | $17$ | $24$ | $12$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-4.d.1.2 | $8$ | $18$ | $18$ | $0$ | $0$ |
| 136.216.7-68.c.1.6 | $136$ | $2$ | $2$ | $7$ | $?$ |
| 136.216.7-68.c.1.21 | $136$ | $2$ | $2$ | $7$ | $?$ |