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Properties

Label 272.24.0.o.1
Level $272$
Index $24$
Genus $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $272$ $\SL_2$-level: $16$
Index: $24$ $\PSL_2$-index:$24$
Genus: $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $1^{4}\cdot4\cdot16$ Cusp orbits $1^{2}\cdot2^{2}$
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: 16C0

Level structure

$\GL_2(\Z/272\Z)$-generators: $\begin{bmatrix}14&147\\39&42\end{bmatrix}$, $\begin{bmatrix}22&19\\261&252\end{bmatrix}$, $\begin{bmatrix}91&154\\66&67\end{bmatrix}$, $\begin{bmatrix}104&23\\123&44\end{bmatrix}$, $\begin{bmatrix}127&14\\152&229\end{bmatrix}$, $\begin{bmatrix}195&142\\96&73\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 272.48.0-272.o.1.1, 272.48.0-272.o.1.2, 272.48.0-272.o.1.3, 272.48.0-272.o.1.4, 272.48.0-272.o.1.5, 272.48.0-272.o.1.6, 272.48.0-272.o.1.7, 272.48.0-272.o.1.8, 272.48.0-272.o.1.9, 272.48.0-272.o.1.10, 272.48.0-272.o.1.11, 272.48.0-272.o.1.12, 272.48.0-272.o.1.13, 272.48.0-272.o.1.14, 272.48.0-272.o.1.15, 272.48.0-272.o.1.16, 272.48.0-272.o.1.17, 272.48.0-272.o.1.18, 272.48.0-272.o.1.19, 272.48.0-272.o.1.20, 272.48.0-272.o.1.21, 272.48.0-272.o.1.22, 272.48.0-272.o.1.23, 272.48.0-272.o.1.24, 272.48.0-272.o.1.25, 272.48.0-272.o.1.26, 272.48.0-272.o.1.27, 272.48.0-272.o.1.28, 272.48.0-272.o.1.29, 272.48.0-272.o.1.30, 272.48.0-272.o.1.31, 272.48.0-272.o.1.32
Cyclic 272-isogeny field degree: $36$
Cyclic 272-torsion field degree: $4608$
Full 272-torsion field degree: $80216064$

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
$X_0(8)$ $8$ $2$ $2$ $0$ $0$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
272.48.0.bk.1 $272$ $2$ $2$ $0$
272.48.0.bk.2 $272$ $2$ $2$ $0$
272.48.0.bl.1 $272$ $2$ $2$ $0$
272.48.0.bl.2 $272$ $2$ $2$ $0$
272.48.0.bm.1 $272$ $2$ $2$ $0$
272.48.0.bm.2 $272$ $2$ $2$ $0$
272.48.0.bn.1 $272$ $2$ $2$ $0$
272.48.0.bn.2 $272$ $2$ $2$ $0$
272.48.0.bo.1 $272$ $2$ $2$ $0$
272.48.0.bo.2 $272$ $2$ $2$ $0$
272.48.0.bp.1 $272$ $2$ $2$ $0$
272.48.0.bp.2 $272$ $2$ $2$ $0$
272.48.0.bq.1 $272$ $2$ $2$ $0$
272.48.0.bq.2 $272$ $2$ $2$ $0$
272.48.0.br.1 $272$ $2$ $2$ $0$
272.48.0.br.2 $272$ $2$ $2$ $0$
272.48.1.b.2 $272$ $2$ $2$ $1$
272.48.1.d.1 $272$ $2$ $2$ $1$
272.48.1.h.1 $272$ $2$ $2$ $1$
272.48.1.i.1 $272$ $2$ $2$ $1$
272.48.1.r.1 $272$ $2$ $2$ $1$
272.48.1.s.1 $272$ $2$ $2$ $1$
272.48.1.v.1 $272$ $2$ $2$ $1$
272.48.1.w.1 $272$ $2$ $2$ $1$