Modular curve 312.192.5.hs.2

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

Invariants

Level:	$312$		$\SL_2$-level:	$8$		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	$8^{24}$		Cusp orbits	$2^{4}\cdot4^{2}\cdot8$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 5$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

8A5

Level structure

$\GL_2(\Z/312\Z)$-generators:	$\begin{bmatrix}71&116\\56&31\end{bmatrix}$, $\begin{bmatrix}85&28\\204&193\end{bmatrix}$, $\begin{bmatrix}243&230\\220&49\end{bmatrix}$, $\begin{bmatrix}307&292\\260&231\end{bmatrix}$, $\begin{bmatrix}311&150\\248&301\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	312.384.5-312.hs.2.1, 312.384.5-312.hs.2.2, 312.384.5-312.hs.2.3, 312.384.5-312.hs.2.4, 312.384.5-312.hs.2.5, 312.384.5-312.hs.2.6, 312.384.5-312.hs.2.7, 312.384.5-312.hs.2.8, 312.384.5-312.hs.2.9, 312.384.5-312.hs.2.10, 312.384.5-312.hs.2.11, 312.384.5-312.hs.2.12, 312.384.5-312.hs.2.13, 312.384.5-312.hs.2.14, 312.384.5-312.hs.2.15, 312.384.5-312.hs.2.16
Cyclic 312-isogeny field degree:	$112$
Cyclic 312-torsion field degree:	$5376$
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.w.2	$24$	$2$	$2$	$1$	$1$
104.96.1.x.1	$104$	$2$	$2$	$1$	$?$
312.96.1.bq.2	$312$	$2$	$2$	$1$	$?$
312.96.3.bv.1	$312$	$2$	$2$	$3$	$?$
312.96.3.by.2	$312$	$2$	$2$	$3$	$?$
312.96.3.cd.1	$312$	$2$	$2$	$3$	$?$
312.96.3.cx.1	$312$	$2$	$2$	$3$	$?$