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 | $2^{2}\cdot4^{2}\cdot34^{2}\cdot68^{2}$ | 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: | $\begin{bmatrix}21&0\\94&45\end{bmatrix}$, $\begin{bmatrix}23&68\\120&39\end{bmatrix}$, $\begin{bmatrix}71&0\\31&81\end{bmatrix}$, $\begin{bmatrix}83&0\\77&93\end{bmatrix}$, $\begin{bmatrix}121&0\\132&107\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 136.216.15.h.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 |
---|---|---|---|---|---|
8.24.0-8.d.1.4 | $8$ | $18$ | $18$ | $0$ | $0$ |
$X_0(17)$ | $17$ | $24$ | $12$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.24.0-8.d.1.4 | $8$ | $18$ | $18$ | $0$ | $0$ |
136.216.7-68.c.1.4 | $136$ | $2$ | $2$ | $7$ | $?$ |
136.216.7-68.c.1.6 | $136$ | $2$ | $2$ | $7$ | $?$ |