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 | $2^{2}\cdot4$ | ||
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: | $\begin{bmatrix}1&90\\41&59\end{bmatrix}$, $\begin{bmatrix}9&94\\73&27\end{bmatrix}$, $\begin{bmatrix}23&98\\47&101\end{bmatrix}$, $\begin{bmatrix}71&46\\64&29\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 272.96.1-136.ev.1.1, 272.96.1-136.ev.1.2, 272.96.1-136.ev.1.3, 272.96.1-136.ev.1.4, 272.96.1-136.ev.1.5, 272.96.1-136.ev.1.6, 272.96.1-136.ev.1.7, 272.96.1-136.ev.1.8 |
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 |
---|---|---|---|---|---|---|
8.24.1.r.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
68.24.0.h.1 | $68$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
136.24.0.cc.1 | $136$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
136.24.0.dj.1 | $136$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
136.24.0.eb.1 | $136$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
136.24.1.y.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
136.24.1.br.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 |
---|---|---|---|---|---|---|
136.96.3.by.1 | $136$ | $2$ | $2$ | $3$ | $?$ | not computed |
136.96.3.bz.1 | $136$ | $2$ | $2$ | $3$ | $?$ | not computed |