Modular curve 84.36.2.b.1

• Label: 84.36.2.b.1
• Level: $84$
• Index: $36$
• Genus: $2$
• Cusps: $4$
• $\Q$-cusps: $2$

Invariants

Level:	$84$		$\SL_2$-level:	$12$		Newform level:	$1$
Index:	$36$		$\PSL_2$-index:	$36$
Genus:	$2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (of which $2$ are rational)		Cusp widths	$6^{2}\cdot12^{2}$		Cusp orbits	$1^{2}\cdot2$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

12B2

Level structure

$\GL_2(\Z/84\Z)$-generators:	$\begin{bmatrix}5&50\\0&55\end{bmatrix}$, $\begin{bmatrix}15&58\\64&55\end{bmatrix}$, $\begin{bmatrix}29&54\\36&77\end{bmatrix}$, $\begin{bmatrix}61&66\\82&5\end{bmatrix}$, $\begin{bmatrix}67&20\\68&47\end{bmatrix}$, $\begin{bmatrix}79&0\\82&47\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	84.72.2-84.b.1.1, 84.72.2-84.b.1.2, 84.72.2-84.b.1.3, 84.72.2-84.b.1.4, 84.72.2-84.b.1.5, 84.72.2-84.b.1.6, 84.72.2-84.b.1.7, 84.72.2-84.b.1.8, 84.72.2-84.b.1.9, 84.72.2-84.b.1.10, 84.72.2-84.b.1.11, 84.72.2-84.b.1.12, 84.72.2-84.b.1.13, 84.72.2-84.b.1.14, 84.72.2-84.b.1.15, 84.72.2-84.b.1.16, 168.72.2-84.b.1.1, 168.72.2-84.b.1.2, 168.72.2-84.b.1.3, 168.72.2-84.b.1.4, 168.72.2-84.b.1.5, 168.72.2-84.b.1.6, 168.72.2-84.b.1.7, 168.72.2-84.b.1.8, 168.72.2-84.b.1.9, 168.72.2-84.b.1.10, 168.72.2-84.b.1.11, 168.72.2-84.b.1.12, 168.72.2-84.b.1.13, 168.72.2-84.b.1.14, 168.72.2-84.b.1.15, 168.72.2-84.b.1.16
Cyclic 84-isogeny field degree:	$64$
Cyclic 84-torsion field degree:	$1536$
Full 84-torsion field degree:	$258048$

Rational points

This modular curve has 2 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
$X_{\mathrm{ns}}^+(3)$	$3$	$12$	$12$	$0$	$0$
28.12.0.b.1	$28$	$3$	$3$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
6.18.1.a.1	$6$	$2$	$2$	$1$	$0$
28.12.0.b.1	$28$	$3$	$3$	$0$	$0$
84.18.0.k.1	$84$	$2$	$2$	$0$	$?$
84.18.1.c.1	$84$	$2$	$2$	$1$	$?$

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

Covering curve	Level	Index	Degree	Genus
84.72.3.b.1	$84$	$2$	$2$	$3$
84.72.3.d.1	$84$	$2$	$2$	$3$
84.72.3.v.1	$84$	$2$	$2$	$3$
84.72.3.x.1	$84$	$2$	$2$	$3$
84.72.3.bl.1	$84$	$2$	$2$	$3$
84.72.3.bn.1	$84$	$2$	$2$	$3$
84.72.3.bq.1	$84$	$2$	$2$	$3$
84.72.3.br.1	$84$	$2$	$2$	$3$
84.72.4.b.1	$84$	$2$	$2$	$4$
84.72.4.c.1	$84$	$2$	$2$	$4$
84.72.4.h.1	$84$	$2$	$2$	$4$
84.72.4.j.1	$84$	$2$	$2$	$4$
84.72.4.w.1	$84$	$2$	$2$	$4$
84.72.4.x.1	$84$	$2$	$2$	$4$
84.72.4.be.1	$84$	$2$	$2$	$4$
84.72.4.bf.1	$84$	$2$	$2$	$4$
84.288.21.f.1	$84$	$8$	$8$	$21$
168.72.3.e.1	$168$	$2$	$2$	$3$
168.72.3.k.1	$168$	$2$	$2$	$3$
168.72.3.cm.1	$168$	$2$	$2$	$3$
168.72.3.cs.1	$168$	$2$	$2$	$3$
168.72.3.ei.1	$168$	$2$	$2$	$3$
168.72.3.eo.1	$168$	$2$	$2$	$3$
168.72.3.ew.1	$168$	$2$	$2$	$3$
168.72.3.ez.1	$168$	$2$	$2$	$3$
168.72.4.d.1	$168$	$2$	$2$	$4$
168.72.4.g.1	$168$	$2$	$2$	$4$
168.72.4.s.1	$168$	$2$	$2$	$4$
168.72.4.y.1	$168$	$2$	$2$	$4$
168.72.4.cw.1	$168$	$2$	$2$	$4$
168.72.4.cz.1	$168$	$2$	$2$	$4$
168.72.4.du.1	$168$	$2$	$2$	$4$
168.72.4.dx.1	$168$	$2$	$2$	$4$
252.108.8.b.1	$252$	$3$	$3$	$8$
252.324.22.f.1	$252$	$9$	$9$	$22$