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 | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | 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: | 16H0 |
Level structure
| $\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}75&142\\60&45\end{bmatrix}$, $\begin{bmatrix}109&270\\106&265\end{bmatrix}$, $\begin{bmatrix}236&175\\65&146\end{bmatrix}$, $\begin{bmatrix}246&75\\65&0\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 272.48.0.bk.1 for the level structure with $-I$) |
| Cyclic 272-isogeny field degree: | $36$ |
| Cyclic 272-torsion field degree: | $1152$ |
| 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 |
|---|---|---|---|---|---|
| 16.48.0-16.e.1.1 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 136.48.0-136.ca.1.1 | $136$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-16.e.1.13 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-272.o.1.1 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-272.o.1.18 | $272$ | $2$ | $2$ | $0$ | $?$ |
| 272.48.0-136.ca.1.13 | $272$ | $2$ | $2$ | $0$ | $?$ |
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.p.2.6 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.u.2.1 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bg.2.6 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.bv.2.5 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.cj.2.5 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.cm.2.1 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.cy.2.5 | $272$ | $2$ | $2$ | $1$ |
| 272.192.1-272.dd.2.3 | $272$ | $2$ | $2$ | $1$ |