Invariants
| Level: | $272$ | $\SL_2$-level: | $16$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $2 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 14 }{2}$ | ||||||
| Cusps: | $14$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{4}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16I2 |
Level structure
| $\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}65&20\\268&199\end{bmatrix}$, $\begin{bmatrix}73&172\\132&103\end{bmatrix}$, $\begin{bmatrix}137&92\\104&107\end{bmatrix}$, $\begin{bmatrix}161&216\\256&189\end{bmatrix}$, $\begin{bmatrix}177&220\\20&81\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 272.96.2.a.1 for the level structure with $-I$) |
| Cyclic 272-isogeny field degree: | $72$ |
| Cyclic 272-torsion field degree: | $2304$ |
| Full 272-torsion field degree: | $10027008$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.96.0-8.c.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 272.96.0-8.c.1.2 | $272$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 272.384.5-272.a.1.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.a.2.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.e.1.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.e.2.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.i.1.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.i.2.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.j.1.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.j.2.5 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.cx.1.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.cx.2.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.da.1.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.da.3.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.em.1.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.em.2.5 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.eq.1.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.eq.2.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.7-272.c.1.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.e.1.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.g.1.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.i.1.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.bw.1.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.by.1.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.cm.1.1 | $272$ | $2$ | $2$ | $7$ |
| 272.384.7-272.co.1.1 | $272$ | $2$ | $2$ | $7$ |