Modular curve 84.24.0.e.1

• Label: 84.24.0.e.1
• Level: $84$
• Index: $24$
• Genus: $0$
• Cusps: $6$
• $\Q$-cusps: $0$

Invariants

Level:	$84$		$\SL_2$-level:	$4$
Index:	$24$		$\PSL_2$-index:	$24$
Genus:	$0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (none of which are rational)		Cusp widths	$4^{6}$		Cusp orbits	$2^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
$\Q$-gonality:	$1 \le \gamma \le 2$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

4G0

Level structure

$\GL_2(\Z/84\Z)$-generators:	$\begin{bmatrix}3&34\\16&9\end{bmatrix}$, $\begin{bmatrix}29&58\\12&19\end{bmatrix}$, $\begin{bmatrix}53&54\\78&23\end{bmatrix}$, $\begin{bmatrix}71&2\\2&31\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	84.48.0-84.e.1.1, 84.48.0-84.e.1.2, 84.48.0-84.e.1.3, 84.48.0-84.e.1.4, 84.48.0-84.e.1.5, 84.48.0-84.e.1.6, 84.48.0-84.e.1.7, 84.48.0-84.e.1.8, 168.48.0-84.e.1.1, 168.48.0-84.e.1.2, 168.48.0-84.e.1.3, 168.48.0-84.e.1.4, 168.48.0-84.e.1.5, 168.48.0-84.e.1.6, 168.48.0-84.e.1.7, 168.48.0-84.e.1.8
Cyclic 84-isogeny field degree:	$64$
Cyclic 84-torsion field degree:	$1536$
Full 84-torsion field degree:	$387072$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

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
12.12.0.a.1	$12$	$2$	$2$	$0$	$0$
28.12.0.b.1	$28$	$2$	$2$	$0$	$0$
84.12.0.b.1	$84$	$2$	$2$	$0$	$?$

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

Covering curve	Level	Index	Degree	Genus
84.72.4.h.1	$84$	$3$	$3$	$4$
84.96.3.h.1	$84$	$4$	$4$	$3$
84.192.11.h.1	$84$	$8$	$8$	$11$