Modular curve 312.192.5.hp.1

• Label: 312.192.5.hp.1
• 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	$2^{6}\cdot4\cdot8$
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:	$\begin{bmatrix}25&70\\104&19\end{bmatrix}$, $\begin{bmatrix}107&220\\164&95\end{bmatrix}$, $\begin{bmatrix}161&130\\112&171\end{bmatrix}$, $\begin{bmatrix}191&182\\240&205\end{bmatrix}$, $\begin{bmatrix}291&206\\308&269\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	312.384.5-312.hp.1.1, 312.384.5-312.hp.1.2, 312.384.5-312.hp.1.3, 312.384.5-312.hp.1.4, 312.384.5-312.hp.1.5, 312.384.5-312.hp.1.6, 312.384.5-312.hp.1.7, 312.384.5-312.hp.1.8, 312.384.5-312.hp.1.9, 312.384.5-312.hp.1.10, 312.384.5-312.hp.1.11, 312.384.5-312.hp.1.12, 312.384.5-312.hp.1.13, 312.384.5-312.hp.1.14, 312.384.5-312.hp.1.15, 312.384.5-312.hp.1.16
Cyclic 312-isogeny field degree:	$112$
Cyclic 312-torsion field degree:	$5376$
Full 312-torsion field degree:	$10063872$

Rational points

This modular curve has no $\Q_p$ points for $p=5$, and therefore no rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
8.96.1.f.2	$8$	$2$	$2$	$1$	$0$
312.96.1.br.2	$312$	$2$	$2$	$1$	$?$
312.96.1.cy.1	$312$	$2$	$2$	$1$	$?$
312.96.3.bp.2	$312$	$2$	$2$	$3$	$?$
312.96.3.bq.1	$312$	$2$	$2$	$3$	$?$
312.96.3.ce.2	$312$	$2$	$2$	$3$	$?$
312.96.3.cw.1	$312$	$2$	$2$	$3$	$?$