Invariants
Level: | $136$ | $\SL_2$-level: | $8$ | Newform level: | $1$ | ||
Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{4}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G1 |
Level structure
$\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}7&54\\12&11\end{bmatrix}$, $\begin{bmatrix}17&108\\88&87\end{bmatrix}$, $\begin{bmatrix}39&98\\12&113\end{bmatrix}$, $\begin{bmatrix}59&84\\92&11\end{bmatrix}$, $\begin{bmatrix}59&106\\56&129\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 136.96.1-136.bh.2.1, 136.96.1-136.bh.2.2, 136.96.1-136.bh.2.3, 136.96.1-136.bh.2.4, 136.96.1-136.bh.2.5, 136.96.1-136.bh.2.6, 136.96.1-136.bh.2.7, 136.96.1-136.bh.2.8, 136.96.1-136.bh.2.9, 136.96.1-136.bh.2.10, 136.96.1-136.bh.2.11, 136.96.1-136.bh.2.12, 136.96.1-136.bh.2.13, 136.96.1-136.bh.2.14, 136.96.1-136.bh.2.15, 136.96.1-136.bh.2.16 |
Cyclic 136-isogeny field degree: | $36$ |
Cyclic 136-torsion field degree: | $2304$ |
Full 136-torsion field degree: | $2506752$ |
Jacobian
Conductor: | $?$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | not computed |
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 | Kernel decomposition |
---|---|---|---|---|---|---|
8.24.0.d.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
136.24.0.i.2 | $136$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
136.24.1.d.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
136.96.1.a.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
136.96.1.r.2 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
136.96.1.bj.2 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
136.96.1.bn.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
136.96.1.bs.2 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
136.96.1.bw.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
136.96.1.ce.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
136.96.1.cg.2 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |