Invariants
| Level: | $272$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
Level structure
| $\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}31&248\\165&117\end{bmatrix}$, $\begin{bmatrix}189&48\\269&119\end{bmatrix}$, $\begin{bmatrix}217&32\\172&213\end{bmatrix}$, $\begin{bmatrix}239&24\\214&231\end{bmatrix}$, $\begin{bmatrix}251&40\\11&137\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 136.48.0.bf.2 for the level structure with $-I$) |
| Cyclic 272-isogeny field degree: | $36$ |
| Cyclic 272-torsion field degree: | $2304$ |
| Full 272-torsion field degree: | $20054016$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.48.0-8.q.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 272.48.0-8.q.1.5 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-136.ca.2.7 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-136.ca.2.14 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-136.cb.1.11 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-136.cb.1.14 | $272$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 272.192.1-272.bf.2.6 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bg.1.2 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bh.1.6 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bk.2.1 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bl.2.6 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bo.1.1 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bp.1.6 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bq.2.2 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-136.cj.1.4 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-136.ck.1.8 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-136.cl.1.4 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-136.cm.2.8 | $272$ | $2$ | $2$ | $1$ |
| 272.192.3-272.gs.1.4 | $272$ | $2$ | $2$ | $3$ |
| 272.192.3-272.gu.2.8 | $272$ | $2$ | $2$ | $3$ |
| 272.192.3-272.ha.2.8 | $272$ | $2$ | $2$ | $3$ |
| 272.192.3-272.he.2.8 | $272$ | $2$ | $2$ | $3$ |