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: | $\begin{bmatrix}8&67\\13&150\end{bmatrix}$, $\begin{bmatrix}35&210\\138&259\end{bmatrix}$, $\begin{bmatrix}47&234\\2&263\end{bmatrix}$, $\begin{bmatrix}239&22\\116&105\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 272.48.0.bs.1 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.1.15 | $16$ | $2$ | $2$ | $0$ | $0$ |
136.48.0-136.ca.2.2 | $136$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-16.e.1.7 | $272$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-272.p.1.23 | $272$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-272.p.1.29 | $272$ | $2$ | $2$ | $0$ | $?$ |
272.48.0-136.ca.2.2 | $272$ | $2$ | $2$ | $0$ | $?$ |
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.m.1.7 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.y.2.8 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.bf.2.7 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.bz.2.8 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.ce.2.4 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.cr.2.10 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.cv.1.6 | $272$ | $2$ | $2$ | $1$ |
272.192.1-272.dg.1.5 | $272$ | $2$ | $2$ | $1$ |