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: | $10$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $2^{3}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16H0 |
Level structure
$\GL_2(\Z/272\Z)$-generators: | $\begin{bmatrix}27&196\\264&15\end{bmatrix}$, $\begin{bmatrix}105&226\\240&107\end{bmatrix}$, $\begin{bmatrix}112&31\\35&124\end{bmatrix}$, $\begin{bmatrix}243&248\\132&239\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 272.48.0.ch.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 real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
16.48.0-16.h.1.14 | $16$ | $2$ | $2$ | $0$ | $0$ |
136.48.0-136.cb.2.5 | $136$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-16.h.1.3 | $272$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-272.n.2.2 | $272$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-272.n.2.4 | $272$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-136.cb.2.11 | $272$ | $2$ | $2$ | $0$ | $?$ |
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.c.1.3 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.bd.1.3 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.bl.1.2 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.cc.1.1 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.dv.1.1 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.ea.1.1 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.em.1.1 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.ep.1.2 | $272$ | $2$ | $2$ | $1$ |