Invariants
| Level: | $272$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-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: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}63&8\\34&93\end{bmatrix}$, $\begin{bmatrix}99&88\\191&199\end{bmatrix}$, $\begin{bmatrix}163&232\\83&253\end{bmatrix}$, $\begin{bmatrix}185&176\\5&111\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 136.48.0.be.1 for the level structure with $-I$) |
| Cyclic 272-isogeny field degree: | $36$ |
| Cyclic 272-torsion field degree: | $2304$ |
| Full 272-torsion field degree: | $20054016$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
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 |
|---|---|---|---|---|---|
| 16.48.0-8.k.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 272.48.0-8.k.1.1 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-136.ca.2.7 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-136.ca.2.12 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-136.cb.2.9 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-136.cb.2.16 | $272$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 272.192.1-272.s.2.3 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.t.1.2 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.u.1.1 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.x.1.7 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.y.2.4 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bb.2.1 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bc.1.9 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bd.1.5 | $272$ | $2$ | $2$ | $1$ |