Invariants
| Level: | $272$ | $\SL_2$-level: | $16$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{12}\cdot16^{4}$ | Cusp orbits | $2^{8}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16O5 |
Level structure
| $\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}13&72\\256&113\end{bmatrix}$, $\begin{bmatrix}47&124\\192&105\end{bmatrix}$, $\begin{bmatrix}57&152\\172&17\end{bmatrix}$, $\begin{bmatrix}131&44\\120&193\end{bmatrix}$, $\begin{bmatrix}183&228\\244&257\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 272.192.5.cz.2 for the level structure with $-I$) |
| Cyclic 272-isogeny field degree: | $36$ |
| Cyclic 272-torsion field degree: | $1152$ |
| Full 272-torsion field degree: | $5013504$ |
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 |
|---|---|---|---|---|---|
| 8.192.1-8.g.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ |
| 272.192.1-8.g.2.8 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.192.2-272.b.1.2 | $272$ | $2$ | $2$ | $2$ | $?$ |
| 272.192.2-272.b.1.25 | $272$ | $2$ | $2$ | $2$ | $?$ |
| 272.192.2-272.k.1.1 | $272$ | $2$ | $2$ | $2$ | $?$ |
| 272.192.2-272.k.1.21 | $272$ | $2$ | $2$ | $2$ | $?$ |