Invariants
Level: | $136$ | $\SL_2$-level: | $136$ | Newform level: | $2312$ | ||
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: | $\begin{bmatrix}3&68\\28&129\end{bmatrix}$, $\begin{bmatrix}73&68\\48&61\end{bmatrix}$, $\begin{bmatrix}89&68\\89&63\end{bmatrix}$, $\begin{bmatrix}117&68\\15&113\end{bmatrix}$, $\begin{bmatrix}125&0\\38&37\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 68.216.15.o.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
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
136.24.0-68.g.1.6 | $136$ | $18$ | $18$ | $0$ | $?$ |
136.216.7-68.c.1.14 | $136$ | $2$ | $2$ | $7$ | $?$ |
136.216.7-68.c.1.21 | $136$ | $2$ | $2$ | $7$ | $?$ |