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: | $\begin{bmatrix}23&176\\158&81\end{bmatrix}$, $\begin{bmatrix}59&168\\78&259\end{bmatrix}$, $\begin{bmatrix}99&120\\263&11\end{bmatrix}$, $\begin{bmatrix}263&112\\182&93\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 272.48.0.be.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 |
---|---|---|---|---|---|
16.48.0-16.e.2.3 | $16$ | $2$ | $2$ | $0$ | $0$ |
136.48.0-136.bn.1.7 | $136$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-16.e.2.1 | $272$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-272.m.1.1 | $272$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-272.m.1.18 | $272$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-136.bn.1.5 | $272$ | $2$ | $2$ | $0$ | $?$ |
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.cu.1.3 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.cv.2.2 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.dc.2.2 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.dd.1.8 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.ec.2.2 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.ed.1.4 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.ek.1.3 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.el.2.2 | $272$ | $2$ | $2$ | $1$ |