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: | $12$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot16^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16H3 |
Level structure
| $\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}21&156\\228&43\end{bmatrix}$, $\begin{bmatrix}81&156\\4&267\end{bmatrix}$, $\begin{bmatrix}165&84\\20&23\end{bmatrix}$, $\begin{bmatrix}169&104\\176&121\end{bmatrix}$, $\begin{bmatrix}209&248\\140&193\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 272.96.3.cb.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 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 |
|---|---|---|---|---|---|
| 8.96.1-8.h.1.2 | $8$ | $2$ | $2$ | $1$ | $0$ |
| 272.96.1-272.a.2.1 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.96.1-272.a.2.20 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.96.1-272.b.1.1 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.96.1-272.b.1.16 | $272$ | $2$ | $2$ | $1$ | $?$ |
| 272.96.1-8.h.1.6 | $272$ | $2$ | $2$ | $1$ | $?$ |
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.bv.2.3 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.bv.3.4 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.bz.1.1 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.bz.4.2 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.cp.1.2 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.cp.3.11 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.db.1.2 | $272$ | $2$ | $2$ | $5$ |
| 272.384.5-272.db.4.6 | $272$ | $2$ | $2$ | $5$ |
| 272.384.9-272.hl.1.2 | $272$ | $2$ | $2$ | $9$ |
| 272.384.9-272.hl.4.2 | $272$ | $2$ | $2$ | $9$ |
| 272.384.9-272.ho.1.2 | $272$ | $2$ | $2$ | $9$ |
| 272.384.9-272.ho.4.1 | $272$ | $2$ | $2$ | $9$ |
| 272.384.9-272.hp.1.2 | $272$ | $2$ | $2$ | $9$ |
| 272.384.9-272.hp.2.2 | $272$ | $2$ | $2$ | $9$ |
| 272.384.9-272.hq.1.1 | $272$ | $2$ | $2$ | $9$ |
| 272.384.9-272.hq.3.3 | $272$ | $2$ | $2$ | $9$ |