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: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G0 |
Level structure
$\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}87&92\\21&47\end{bmatrix}$, $\begin{bmatrix}103&56\\10&61\end{bmatrix}$, $\begin{bmatrix}131&84\\86&133\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 136.24.0.bo.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 real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.24.0-8.m.1.8 | $8$ | $2$ | $2$ | $0$ | $0$ |
136.24.0-8.m.1.7 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.24.0-136.v.1.4 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.24.0-136.v.1.6 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.24.0-136.z.1.2 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.24.0-136.z.1.7 | $136$ | $2$ | $2$ | $0$ | $?$ |