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 | $4^{2}$ | ||
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: | 8F1 |
Level structure
$\GL_2(\Z/136\Z)$-generators: | $\begin{bmatrix}83&72\\67&81\end{bmatrix}$, $\begin{bmatrix}93&2\\95&99\end{bmatrix}$, $\begin{bmatrix}133&108\\23&71\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 272.96.1-136.hi.1.1, 272.96.1-136.hi.1.2, 272.96.1-136.hi.1.3, 272.96.1-136.hi.1.4, 272.96.1-136.hi.1.5, 272.96.1-136.hi.1.6 |
Cyclic 136-isogeny field degree: | $72$ |
Cyclic 136-torsion field degree: | $4608$ |
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.bc.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
136.24.0.cc.1 | $136$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
136.24.0.ck.1 | $136$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
136.24.0.ew.1 | $136$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
136.24.1.ca.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
136.24.1.cs.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
136.24.1.dk.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 |
---|---|---|---|---|---|---|
272.96.3.vd.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
272.96.3.ve.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
272.96.3.vf.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
272.96.3.vg.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |