Invariants
Level: | $136$ | $\SL_2$-level: | $8$ | ||||
Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8O0 |
Level structure
$\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}11&92\\60&45\end{bmatrix}$, $\begin{bmatrix}17&60\\28&123\end{bmatrix}$, $\begin{bmatrix}47&12\\102&71\end{bmatrix}$, $\begin{bmatrix}65&76\\38&129\end{bmatrix}$, $\begin{bmatrix}67&52\\26&13\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 136.96.0-136.u.2.1, 136.96.0-136.u.2.2, 136.96.0-136.u.2.3, 136.96.0-136.u.2.4, 136.96.0-136.u.2.5, 136.96.0-136.u.2.6, 136.96.0-136.u.2.7, 136.96.0-136.u.2.8, 136.96.0-136.u.2.9, 136.96.0-136.u.2.10, 136.96.0-136.u.2.11, 136.96.0-136.u.2.12, 136.96.0-136.u.2.13, 136.96.0-136.u.2.14, 136.96.0-136.u.2.15, 136.96.0-136.u.2.16 |
Cyclic 136-isogeny field degree: | $36$ |
Cyclic 136-torsion field degree: | $2304$ |
Full 136-torsion field degree: | $2506752$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.24.0.d.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
136.24.0.h.2 | $136$ | $2$ | $2$ | $0$ | $?$ |
136.24.0.m.1 | $136$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
136.96.1.b.1 | $136$ | $2$ | $2$ | $1$ |
136.96.1.c.1 | $136$ | $2$ | $2$ | $1$ |
136.96.1.h.2 | $136$ | $2$ | $2$ | $1$ |
136.96.1.i.2 | $136$ | $2$ | $2$ | $1$ |
136.96.1.bi.2 | $136$ | $2$ | $2$ | $1$ |
136.96.1.bj.2 | $136$ | $2$ | $2$ | $1$ |
136.96.1.bk.1 | $136$ | $2$ | $2$ | $1$ |
136.96.1.bl.1 | $136$ | $2$ | $2$ | $1$ |