Invariants
| Level: | $272$ | $\SL_2$-level: | $16$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{8}\cdot16^{4}$ | Cusp orbits | $2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16H3 |
Level structure
| $\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}9&4\\74&47\end{bmatrix}$, $\begin{bmatrix}21&128\\230&257\end{bmatrix}$, $\begin{bmatrix}81&28\\170&143\end{bmatrix}$, $\begin{bmatrix}161&232\\270&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 272.96.3.ch.1 for the level structure with $-I$) |
| Cyclic 272-isogeny field degree: | $36$ |
| Cyclic 272-torsion field degree: | $4608$ |
| Full 272-torsion field degree: | $10027008$ |
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 |
|---|---|---|---|---|---|
| 8.96.1-8.p.1.2 | $8$ | $2$ | $2$ | $1$ | $0$ |
| 272.96.1-272.a.1.1 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.96.1-272.a.1.20 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.96.1-272.a.2.1 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.96.1-272.a.2.24 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.96.1-8.p.1.3 | $272$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 272.384.5-272.dg.1.3 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.dg.2.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.di.1.2 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.di.2.7 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.dk.1.2 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.dk.2.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.dm.1.3 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.dm.2.4 | $272$ | $2$ | $2$ | $5$ |