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Properties

Label 112.24.0.e.1
Level $112$
Index $24$
Genus $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $112$ $\SL_2$-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

$\GL_2(\Z/112\Z)$-generators: $\begin{bmatrix}12&25\\27&82\end{bmatrix}$, $\begin{bmatrix}29&32\\44&97\end{bmatrix}$, $\begin{bmatrix}48&97\\43&70\end{bmatrix}$, $\begin{bmatrix}52&87\\71&52\end{bmatrix}$, $\begin{bmatrix}54&51\\83&94\end{bmatrix}$, $\begin{bmatrix}108&17\\99&10\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 112.48.0-112.e.1.1, 112.48.0-112.e.1.2, 112.48.0-112.e.1.3, 112.48.0-112.e.1.4, 112.48.0-112.e.1.5, 112.48.0-112.e.1.6, 112.48.0-112.e.1.7, 112.48.0-112.e.1.8, 112.48.0-112.e.1.9, 112.48.0-112.e.1.10, 112.48.0-112.e.1.11, 112.48.0-112.e.1.12, 112.48.0-112.e.1.13, 112.48.0-112.e.1.14, 112.48.0-112.e.1.15, 112.48.0-112.e.1.16, 112.48.0-112.e.1.17, 112.48.0-112.e.1.18, 112.48.0-112.e.1.19, 112.48.0-112.e.1.20, 112.48.0-112.e.1.21, 112.48.0-112.e.1.22, 112.48.0-112.e.1.23, 112.48.0-112.e.1.24, 112.48.0-112.e.1.25, 112.48.0-112.e.1.26, 112.48.0-112.e.1.27, 112.48.0-112.e.1.28, 112.48.0-112.e.1.29, 112.48.0-112.e.1.30, 112.48.0-112.e.1.31, 112.48.0-112.e.1.32
Cyclic 112-isogeny field degree: $16$
Cyclic 112-torsion field degree: $768$
Full 112-torsion field degree: $2064384$

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
112.48.0.d.1 $112$ $2$ $2$ $0$
112.48.0.e.1 $112$ $2$ $2$ $0$
112.48.0.j.1 $112$ $2$ $2$ $0$
112.48.0.l.1 $112$ $2$ $2$ $0$
112.48.0.v.1 $112$ $2$ $2$ $0$
112.48.0.w.2 $112$ $2$ $2$ $0$
112.48.0.y.2 $112$ $2$ $2$ $0$
112.48.0.bb.1 $112$ $2$ $2$ $0$
112.48.0.be.1 $112$ $2$ $2$ $0$
112.48.0.bf.1 $112$ $2$ $2$ $0$
112.48.0.bm.1 $112$ $2$ $2$ $0$
112.48.0.bn.1 $112$ $2$ $2$ $0$
112.48.0.bs.1 $112$ $2$ $2$ $0$
112.48.0.bt.1 $112$ $2$ $2$ $0$
112.48.0.bw.1 $112$ $2$ $2$ $0$
112.48.0.bx.1 $112$ $2$ $2$ $0$
112.48.1.bg.1 $112$ $2$ $2$ $1$
112.48.1.bh.2 $112$ $2$ $2$ $1$
112.48.1.bk.2 $112$ $2$ $2$ $1$
112.48.1.bl.1 $112$ $2$ $2$ $1$
112.48.1.bq.1 $112$ $2$ $2$ $1$
112.48.1.br.2 $112$ $2$ $2$ $1$
112.48.1.by.2 $112$ $2$ $2$ $1$
112.48.1.bz.1 $112$ $2$ $2$ $1$
112.192.11.q.1 $112$ $8$ $8$ $11$