Modular curve 312.192.5.ko.2

• Label: 312.192.5.ko.2
• Level: $312$
• Index: $192$
• Genus: $5$
• Cusps: $24$
• $\Q$-cusps: $0$

Invariants

Level:	$312$		$\SL_2$-level:	$12$		Newform level:	$1$
Index:	$192$		$\PSL_2$-index:	$192$
Genus:	$5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps:	$24$ (none of which are rational)		Cusp widths	$4^{12}\cdot12^{12}$		Cusp orbits	$2^{2}\cdot4^{5}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 5$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

12E5

Level structure

$\GL_2(\Z/312\Z)$-generators:	$\begin{bmatrix}31&206\\18&227\end{bmatrix}$, $\begin{bmatrix}69&172\\220&75\end{bmatrix}$, $\begin{bmatrix}101&18\\44&79\end{bmatrix}$, $\begin{bmatrix}191&150\\234&191\end{bmatrix}$, $\begin{bmatrix}287&58\\0&37\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	312.384.5-312.ko.2.1, 312.384.5-312.ko.2.2, 312.384.5-312.ko.2.3, 312.384.5-312.ko.2.4, 312.384.5-312.ko.2.5, 312.384.5-312.ko.2.6, 312.384.5-312.ko.2.7, 312.384.5-312.ko.2.8, 312.384.5-312.ko.2.9, 312.384.5-312.ko.2.10, 312.384.5-312.ko.2.11, 312.384.5-312.ko.2.12, 312.384.5-312.ko.2.13, 312.384.5-312.ko.2.14, 312.384.5-312.ko.2.15, 312.384.5-312.ko.2.16
Cyclic 312-isogeny field degree:	$56$
Cyclic 312-torsion field degree:	$2688$
Full 312-torsion field degree:	$10063872$

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
24.96.1.cp.2	$24$	$2$	$2$	$1$	$1$
156.96.3.o.1	$156$	$2$	$2$	$3$	$?$
312.96.1.ln.2	$312$	$2$	$2$	$1$	$?$
312.96.1.lv.4	$312$	$2$	$2$	$1$	$?$
312.96.3.ej.1	$312$	$2$	$2$	$3$	$?$
312.96.3.fj.1	$312$	$2$	$2$	$3$	$?$
312.96.3.fv.2	$312$	$2$	$2$	$3$	$?$