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 | $2^{8}\cdot16^{2}$ | 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: | 16G0 |
Level structure
| $\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}49&260\\10&55\end{bmatrix}$, $\begin{bmatrix}69&48\\110&73\end{bmatrix}$, $\begin{bmatrix}101&0\\248&5\end{bmatrix}$, $\begin{bmatrix}117&224\\120&225\end{bmatrix}$, $\begin{bmatrix}129&216\\74&61\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 272.48.0.f.1 for the level structure with $-I$) |
| Cyclic 272-isogeny field degree: | $36$ |
| Cyclic 272-torsion field degree: | $4608$ |
| 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 |
|---|---|---|---|---|---|
| 8.48.0-8.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 272.48.0-8.i.1.3 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-272.m.2.8 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-272.m.2.25 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-272.n.1.9 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-272.n.1.24 | $272$ | $2$ | $2$ | $0$ | $?$ |
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.a.1.2 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.b.2.4 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.c.2.12 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.d.2.4 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.e.2.3 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.f.1.2 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.g.1.4 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.h.1.2 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.i.2.8 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.j.2.2 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.k.1.1 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.l.1.2 | $272$ | $2$ | $2$ | $1$ |
| 272.192.3-272.cc.1.1 | $272$ | $2$ | $2$ | $3$ |
| 272.192.3-272.ce.1.1 | $272$ | $2$ | $2$ | $3$ |
| 272.192.3-272.cj.1.7 | $272$ | $2$ | $2$ | $3$ |
| 272.192.3-272.cm.2.5 | $272$ | $2$ | $2$ | $3$ |