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Properties

Label 272.24.0.n.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 $1^{2}\cdot2^{3}\cdot16$ 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: 16D0

Level structure

$\GL_2(\Z/272\Z)$-generators: $\begin{bmatrix}7&232\\188&67\end{bmatrix}$, $\begin{bmatrix}12&111\\91&232\end{bmatrix}$, $\begin{bmatrix}13&144\\26&195\end{bmatrix}$, $\begin{bmatrix}73&96\\144&161\end{bmatrix}$, $\begin{bmatrix}102&191\\185&196\end{bmatrix}$, $\begin{bmatrix}108&185\\205&152\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 272.48.0-272.n.1.1, 272.48.0-272.n.1.2, 272.48.0-272.n.1.3, 272.48.0-272.n.1.4, 272.48.0-272.n.1.5, 272.48.0-272.n.1.6, 272.48.0-272.n.1.7, 272.48.0-272.n.1.8, 272.48.0-272.n.1.9, 272.48.0-272.n.1.10, 272.48.0-272.n.1.11, 272.48.0-272.n.1.12, 272.48.0-272.n.1.13, 272.48.0-272.n.1.14, 272.48.0-272.n.1.15, 272.48.0-272.n.1.16, 272.48.0-272.n.1.17, 272.48.0-272.n.1.18, 272.48.0-272.n.1.19, 272.48.0-272.n.1.20, 272.48.0-272.n.1.21, 272.48.0-272.n.1.22, 272.48.0-272.n.1.23, 272.48.0-272.n.1.24, 272.48.0-272.n.1.25, 272.48.0-272.n.1.26, 272.48.0-272.n.1.27, 272.48.0-272.n.1.28, 272.48.0-272.n.1.29, 272.48.0-272.n.1.30, 272.48.0-272.n.1.31, 272.48.0-272.n.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.f.1 $272$ $2$ $2$ $0$
272.48.0.h.1 $272$ $2$ $2$ $0$
272.48.0.m.1 $272$ $2$ $2$ $0$
272.48.0.n.1 $272$ $2$ $2$ $0$
272.48.0.ba.2 $272$ $2$ $2$ $0$
272.48.0.bd.2 $272$ $2$ $2$ $0$
272.48.0.bf.2 $272$ $2$ $2$ $0$
272.48.0.bg.2 $272$ $2$ $2$ $0$
272.48.0.bo.2 $272$ $2$ $2$ $0$
272.48.0.bp.1 $272$ $2$ $2$ $0$
272.48.0.bw.2 $272$ $2$ $2$ $0$
272.48.0.bx.2 $272$ $2$ $2$ $0$
272.48.0.cc.2 $272$ $2$ $2$ $0$
272.48.0.cd.2 $272$ $2$ $2$ $0$
272.48.0.cg.2 $272$ $2$ $2$ $0$
272.48.0.ch.1 $272$ $2$ $2$ $0$
272.48.1.bi.2 $272$ $2$ $2$ $1$
272.48.1.bj.2 $272$ $2$ $2$ $1$
272.48.1.bm.2 $272$ $2$ $2$ $1$
272.48.1.bn.2 $272$ $2$ $2$ $1$
272.48.1.bu.2 $272$ $2$ $2$ $1$
272.48.1.bv.2 $272$ $2$ $2$ $1$
272.48.1.cc.2 $272$ $2$ $2$ $1$
272.48.1.cd.2 $272$ $2$ $2$ $1$