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: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16J3 |
Level structure
| $\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}35&262\\68&257\end{bmatrix}$, $\begin{bmatrix}79&126\\220&149\end{bmatrix}$, $\begin{bmatrix}121&206\\264&177\end{bmatrix}$, $\begin{bmatrix}175&14\\68&65\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 272.96.3.cf.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 |
|---|---|---|---|---|---|
| 16.96.2-16.d.2.9 | $16$ | $2$ | $2$ | $2$ | $0$ |
| 136.96.0-136.bc.1.1 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 272.96.0-136.bc.1.6 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.96.1-272.a.2.1 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.96.1-272.a.2.19 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.96.2-16.d.2.5 | $272$ | $2$ | $2$ | $2$ | $?$ |
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.bx.2.2 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.bz.1.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.cp.3.11 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.dd.2.8 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.dh.2.7 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.di.2.7 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.dk.2.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.dn.1.1 | $272$ | $2$ | $2$ | $5$ |