Properties

Label 112.384.13-112.g.1.4
Level $112$
Index $384$
Genus $13$
Cusps $8$
$\Q$-cusps $8$

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Invariants

Level: $112$ $\SL_2$-level: $112$ Newform level: $1$
Index: $384$ $\PSL_2$-index:$192$
Genus: $13 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (all of which are rational) Cusp widths $2^{2}\cdot4\cdot14^{2}\cdot16\cdot28\cdot112$ Cusp orbits $1^{8}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $4 \le \gamma \le 13$
$\overline{\Q}$-gonality: $4 \le \gamma \le 13$
Rational cusps: $8$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 112D13

Level structure

$\GL_2(\Z/112\Z)$-generators: $\begin{bmatrix}2&29\\57&30\end{bmatrix}$, $\begin{bmatrix}3&0\\68&47\end{bmatrix}$, $\begin{bmatrix}19&34\\8&73\end{bmatrix}$, $\begin{bmatrix}74&5\\65&70\end{bmatrix}$, $\begin{bmatrix}82&11\\67&82\end{bmatrix}$, $\begin{bmatrix}82&15\\9&4\end{bmatrix}$
Contains $-I$: no $\quad$ (see 112.192.13.g.1 for the level structure with $-I$)
Cyclic 112-isogeny field degree: $2$
Cyclic 112-torsion field degree: $48$
Full 112-torsion field degree: $129024$

Rational points

This modular curve has 8 rational cusps but no known non-cuspidal rational points.

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank
$X_0(7)$ $7$ $48$ $24$ $0$ $0$
16.48.1-16.a.1.10 $16$ $8$ $8$ $1$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
16.48.1-16.a.1.10 $16$ $8$ $8$ $1$ $0$
56.192.5-56.bl.1.47 $56$ $2$ $2$ $5$ $0$
112.192.5-56.bl.1.31 $112$ $2$ $2$ $5$ $?$