Modular curve 312.48.0.z.2

• Label: 312.48.0.z.2
• Level: $312$
• Index: $48$
• Genus: $0$
• Cusps: $10$
• $\Q$-cusps: $0$

Invariants

Level:	$312$		$\SL_2$-level:	$8$
Index:	$48$		$\PSL_2$-index:	$48$
Genus:	$0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps:	$10$ (none of which are rational)		Cusp widths	$4^{8}\cdot8^{2}$		Cusp orbits	$2^{3}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
$\Q$-gonality:	$1 \le \gamma \le 2$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

8N0

Level structure

$\GL_2(\Z/312\Z)$-generators:	$\begin{bmatrix}3&104\\284&93\end{bmatrix}$, $\begin{bmatrix}113&240\\132&149\end{bmatrix}$, $\begin{bmatrix}143&86\\268&257\end{bmatrix}$, $\begin{bmatrix}173&286\\272&113\end{bmatrix}$, $\begin{bmatrix}199&72\\224&95\end{bmatrix}$, $\begin{bmatrix}259&278\\168&283\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	312.96.0-312.z.2.1, 312.96.0-312.z.2.2, 312.96.0-312.z.2.3, 312.96.0-312.z.2.4, 312.96.0-312.z.2.5, 312.96.0-312.z.2.6, 312.96.0-312.z.2.7, 312.96.0-312.z.2.8, 312.96.0-312.z.2.9, 312.96.0-312.z.2.10, 312.96.0-312.z.2.11, 312.96.0-312.z.2.12, 312.96.0-312.z.2.13, 312.96.0-312.z.2.14, 312.96.0-312.z.2.15, 312.96.0-312.z.2.16, 312.96.0-312.z.2.17, 312.96.0-312.z.2.18, 312.96.0-312.z.2.19, 312.96.0-312.z.2.20, 312.96.0-312.z.2.21, 312.96.0-312.z.2.22, 312.96.0-312.z.2.23, 312.96.0-312.z.2.24, 312.96.0-312.z.2.25, 312.96.0-312.z.2.26, 312.96.0-312.z.2.27, 312.96.0-312.z.2.28, 312.96.0-312.z.2.29, 312.96.0-312.z.2.30, 312.96.0-312.z.2.31, 312.96.0-312.z.2.32
Cyclic 312-isogeny field degree:	$112$
Cyclic 312-torsion field degree:	$10752$
Full 312-torsion field degree:	$40255488$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
8.24.0.d.1	$8$	$2$	$2$	$0$	$0$
156.24.0.c.1	$156$	$2$	$2$	$0$	$?$
312.24.0.t.2	$312$	$2$	$2$	$0$	$?$

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

Covering curve	Level	Index	Degree	Genus
312.96.1.c.1	$312$	$2$	$2$	$1$
312.96.1.be.2	$312$	$2$	$2$	$1$
312.96.1.ds.2	$312$	$2$	$2$	$1$
312.96.1.dy.1	$312$	$2$	$2$	$1$
312.96.1.iv.2	$312$	$2$	$2$	$1$
312.96.1.ix.1	$312$	$2$	$2$	$1$
312.96.1.jk.1	$312$	$2$	$2$	$1$
312.96.1.jm.2	$312$	$2$	$2$	$1$
312.96.1.mq.1	$312$	$2$	$2$	$1$
312.96.1.ms.2	$312$	$2$	$2$	$1$
312.96.1.nh.2	$312$	$2$	$2$	$1$
312.96.1.nj.1	$312$	$2$	$2$	$1$
312.96.1.pa.2	$312$	$2$	$2$	$1$
312.96.1.pg.1	$312$	$2$	$2$	$1$
312.96.1.pq.1	$312$	$2$	$2$	$1$
312.96.1.pt.2	$312$	$2$	$2$	$1$
312.144.8.dc.2	$312$	$3$	$3$	$8$
312.192.7.de.2	$312$	$4$	$4$	$7$