Modular curve 312.112.7.c.1

• Label: 312.112.7.c.1
• Level: $312$
• Index: $112$
• Genus: $7$
• Cusps: $4$
• $\Q$-cusps: $4$

Invariants

Level:	$312$		$\SL_2$-level:	$78$		Newform level:	$1$
Index:	$112$		$\PSL_2$-index:	$112$
Genus:	$7 = 1 + \frac{ 112 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (all of which are rational)		Cusp widths	$2\cdot6\cdot26\cdot78$		Cusp orbits	$1^{4}$
Elliptic points:	$0$ of order $2$ and $4$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 7$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 7$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

78C7

Level structure

$\GL_2(\Z/312\Z)$-generators:	$\begin{bmatrix}11&147\\282&305\end{bmatrix}$, $\begin{bmatrix}24&19\\23&7\end{bmatrix}$, $\begin{bmatrix}77&256\\80&45\end{bmatrix}$, $\begin{bmatrix}110&123\\183&50\end{bmatrix}$, $\begin{bmatrix}128&231\\311&139\end{bmatrix}$, $\begin{bmatrix}189&88\\272&109\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	312.224.7-312.c.1.1, 312.224.7-312.c.1.2, 312.224.7-312.c.1.3, 312.224.7-312.c.1.4, 312.224.7-312.c.1.5, 312.224.7-312.c.1.6, 312.224.7-312.c.1.7, 312.224.7-312.c.1.8, 312.224.7-312.c.1.9, 312.224.7-312.c.1.10, 312.224.7-312.c.1.11, 312.224.7-312.c.1.12, 312.224.7-312.c.1.13, 312.224.7-312.c.1.14, 312.224.7-312.c.1.15, 312.224.7-312.c.1.16, 312.224.7-312.c.1.17, 312.224.7-312.c.1.18, 312.224.7-312.c.1.19, 312.224.7-312.c.1.20, 312.224.7-312.c.1.21, 312.224.7-312.c.1.22, 312.224.7-312.c.1.23, 312.224.7-312.c.1.24, 312.224.7-312.c.1.25, 312.224.7-312.c.1.26, 312.224.7-312.c.1.27, 312.224.7-312.c.1.28, 312.224.7-312.c.1.29, 312.224.7-312.c.1.30, 312.224.7-312.c.1.31, 312.224.7-312.c.1.32
Cyclic 312-isogeny field degree:	$12$
Cyclic 312-torsion field degree:	$1152$
Full 312-torsion field degree:	$17252352$

Rational points

This modular curve has 4 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
$X_0(13)$	$13$	$8$	$8$	$0$	$0$
24.8.0.c.1	$24$	$14$	$14$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
24.8.0.c.1	$24$	$14$	$14$	$0$	$0$
$X_0(39)$	$39$	$2$	$2$	$3$	$0$
312.28.1.a.1	$312$	$4$	$4$	$1$	$?$

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

Covering curve	Level	Index	Degree	Genus
312.224.13.bl.1	$312$	$2$	$2$	$13$
312.224.13.bl.2	$312$	$2$	$2$	$13$
312.224.13.bm.1	$312$	$2$	$2$	$13$
312.224.13.bm.2	$312$	$2$	$2$	$13$
312.224.13.bo.1	$312$	$2$	$2$	$13$
312.224.13.bo.2	$312$	$2$	$2$	$13$
312.224.13.bp.1	$312$	$2$	$2$	$13$
312.224.13.bp.2	$312$	$2$	$2$	$13$
312.224.13.cj.1	$312$	$2$	$2$	$13$
312.224.13.cj.2	$312$	$2$	$2$	$13$
312.224.13.ck.1	$312$	$2$	$2$	$13$
312.224.13.ck.2	$312$	$2$	$2$	$13$
312.224.13.cm.1	$312$	$2$	$2$	$13$
312.224.13.cm.2	$312$	$2$	$2$	$13$
312.224.13.cn.1	$312$	$2$	$2$	$13$
312.224.13.cn.2	$312$	$2$	$2$	$13$
312.336.23.pp.1	$312$	$3$	$3$	$23$
312.336.23.sz.1	$312$	$3$	$3$	$23$
312.336.23.sz.2	$312$	$3$	$3$	$23$
312.336.23.td.1	$312$	$3$	$3$	$23$