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 | $1^{2}\cdot2^{3}\cdot16$ | 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: | 16D0 |
Level structure
| $\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}18&215\\231&202\end{bmatrix}$, $\begin{bmatrix}40&95\\31&104\end{bmatrix}$, $\begin{bmatrix}125&262\\266&209\end{bmatrix}$, $\begin{bmatrix}130&169\\165&214\end{bmatrix}$, $\begin{bmatrix}220&71\\137&210\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 272.24.0.m.2 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 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 136.24.0-8.n.1.7 | $136$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 272.96.0-272.f.1.18 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.g.1.10 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.l.1.6 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.n.2.9 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bb.1.5 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bc.1.7 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.be.1.1 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bh.1.1 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bm.2.10 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bn.2.14 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bu.2.12 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.bv.2.10 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.ca.2.12 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.cb.2.18 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.ce.2.10 | $272$ | $2$ | $2$ | $0$ |
| 272.96.0-272.cf.2.14 | $272$ | $2$ | $2$ | $0$ |
| 272.96.1-272.bg.2.14 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.bh.1.6 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.bk.2.6 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.bl.2.20 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.bs.2.10 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.bt.2.12 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.ca.2.14 | $272$ | $2$ | $2$ | $1$ |
| 272.96.1-272.cb.1.10 | $272$ | $2$ | $2$ | $1$ |