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: | $\begin{bmatrix}59&228\\200&169\end{bmatrix}$, $\begin{bmatrix}149&138\\200&161\end{bmatrix}$, $\begin{bmatrix}169&250\\88&193\end{bmatrix}$, $\begin{bmatrix}251&70\\80&225\end{bmatrix}$, $\begin{bmatrix}259&138\\240&149\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 272.96.3.dg.2 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 |
|---|---|---|---|---|---|
| 16.96.1-16.a.2.4 | $16$ | $2$ | $2$ | $1$ | $0$ |
| 136.96.0-136.bd.1.3 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 272.96.0-136.bd.1.15 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.96.1-16.a.2.11 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.96.2-272.f.1.12 | $272$ | $2$ | $2$ | $2$ | $?$ |
| 272.96.2-272.f.1.13 | $272$ | $2$ | $2$ | $2$ | $?$ |
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.co.1.2 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.cq.2.2 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.hj.1.2 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.hk.2.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.7-272.dy.2.3 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.dz.2.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.ed.2.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.ee.2.3 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.ek.2.2 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.el.2.3 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.em.2.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.en.2.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.ey.2.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.ez.2.3 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.fb.2.3 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.fc.2.1 | $272$ | $2$ | $2$ | $7$ |