Invariants
| Level: | $272$ | $\SL_2$-level: | $16$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 8I0 |
Level structure
| $\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}83&226\\238&71\end{bmatrix}$, $\begin{bmatrix}98&181\\133&58\end{bmatrix}$, $\begin{bmatrix}100&37\\201&224\end{bmatrix}$, $\begin{bmatrix}209&256\\82&159\end{bmatrix}$, $\begin{bmatrix}239&86\\208&229\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 136.24.0.cb.1 for the level structure with $-I$) |
| Cyclic 272-isogeny field degree: | $36$ |
| Cyclic 272-torsion field degree: | $2304$ |
| Full 272-torsion field degree: | $40108032$ |
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 |
|---|---|---|---|---|---|
| 16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 272.24.0-8.n.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 |
|---|---|---|---|---|
| 272.96.0-136.bd.2.10 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-136.be.2.4 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-136.bf.2.8 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-136.bh.1.6 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-136.bj.1.7 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-136.bk.1.7 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-136.bm.2.5 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-136.bp.2.7 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bl.2.2 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.br.1.2 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bt.1.2 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bz.1.2 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.cb.1.6 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.cd.1.6 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.cf.2.10 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.ch.1.6 | $272$ | $2$ | $2$ | $0$ |
| 272.96.1-272.bh.1.6 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.bj.2.10 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.bl.1.10 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.bn.1.4 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.br.1.2 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.bx.1.2 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.bz.1.2 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.cf.2.2 | $272$ | $2$ | $2$ | $1$ |