Modular curve 312.224.15.e.1

• Label: 312.224.15.e.1
• Level: $312$
• Index: $224$
• Genus: $15$
• Cusps: $4$
• $\Q$-cusps: $4$

Invariants

Level:	$312$		$\SL_2$-level:	$156$		Newform level:	$1$
Index:	$224$		$\PSL_2$-index:	$224$
Genus:	$15 = 1 + \frac{ 224 }{12} - \frac{ 0 }{4} - \frac{ 8 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (all of which are rational)		Cusp widths	$4\cdot12\cdot52\cdot156$		Cusp orbits	$1^{4}$
Elliptic points:	$0$ of order $2$ and $8$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 15$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 15$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

156B15

Level structure

$\GL_2(\Z/312\Z)$-generators:	$\begin{bmatrix}45&298\\20&115\end{bmatrix}$, $\begin{bmatrix}110&93\\195&203\end{bmatrix}$, $\begin{bmatrix}138&17\\169&233\end{bmatrix}$, $\begin{bmatrix}173&96\\84&185\end{bmatrix}$, $\begin{bmatrix}255&310\\64&267\end{bmatrix}$, $\begin{bmatrix}266&169\\111&103\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	312.448.15-312.e.1.1, 312.448.15-312.e.1.2, 312.448.15-312.e.1.3, 312.448.15-312.e.1.4, 312.448.15-312.e.1.5, 312.448.15-312.e.1.6, 312.448.15-312.e.1.7, 312.448.15-312.e.1.8, 312.448.15-312.e.1.9, 312.448.15-312.e.1.10, 312.448.15-312.e.1.11, 312.448.15-312.e.1.12, 312.448.15-312.e.1.13, 312.448.15-312.e.1.14, 312.448.15-312.e.1.15, 312.448.15-312.e.1.16, 312.448.15-312.e.1.17, 312.448.15-312.e.1.18, 312.448.15-312.e.1.19, 312.448.15-312.e.1.20, 312.448.15-312.e.1.21, 312.448.15-312.e.1.22, 312.448.15-312.e.1.23, 312.448.15-312.e.1.24, 312.448.15-312.e.1.25, 312.448.15-312.e.1.26, 312.448.15-312.e.1.27, 312.448.15-312.e.1.28, 312.448.15-312.e.1.29, 312.448.15-312.e.1.30, 312.448.15-312.e.1.31, 312.448.15-312.e.1.32
Cyclic 312-isogeny field degree:	$12$
Cyclic 312-torsion field degree:	$1152$
Full 312-torsion field degree:	$8626176$

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
$X_0(13)$	$13$	$16$	$16$	$0$	$0$
24.16.0.a.1	$24$	$14$	$14$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
24.16.0.a.1	$24$	$14$	$14$	$0$	$0$
78.112.7.a.1	$78$	$2$	$2$	$7$	$?$