Invariants
| Level: | $136$ | $\SL_2$-level: | $136$ | Newform level: | $1$ | ||
| 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 | $1^{2}\cdot2\cdot8\cdot17^{2}\cdot34\cdot136$ | 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: | 136D15 |
Level structure
| $\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}45&12\\18&39\end{bmatrix}$, $\begin{bmatrix}97&64\\110&51\end{bmatrix}$, $\begin{bmatrix}117&28\\114&99\end{bmatrix}$, $\begin{bmatrix}117&116\\44&121\end{bmatrix}$, $\begin{bmatrix}134&129\\127&0\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 136.216.15.ci.1 for the level structure with $-I$) |
| Cyclic 136-isogeny field degree: | $2$ |
| Cyclic 136-torsion field degree: | $64$ |
| 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 |
|---|---|---|---|---|---|
| 8.24.0-8.m.1.8 | $8$ | $18$ | $18$ | $0$ | $0$ |
| $X_0(17)$ | $17$ | $24$ | $12$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.m.1.8 | $8$ | $18$ | $18$ | $0$ | $0$ |
| 68.216.7-68.c.1.8 | $68$ | $2$ | $2$ | $7$ | $0$ |
| 136.216.7-68.c.1.21 | $136$ | $2$ | $2$ | $7$ | $?$ |