Modular curve 84.48.1.b.1

• Label: 84.48.1.b.1
• Level: $84$
• Index: $48$
• Genus: $1$
• Cusps: $8$
• $\Q$-cusps: $4$

Invariants

Level:	$84$		$\SL_2$-level:	$12$		Newform level:	$1$
Index:	$48$		$\PSL_2$-index:	$48$
Genus:	$1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (of which $4$ are rational)		Cusp widths	$2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$		Cusp orbits	$1^{4}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

12P1

Level structure

$\GL_2(\Z/84\Z)$-generators:	$\begin{bmatrix}45&8\\82&83\end{bmatrix}$, $\begin{bmatrix}47&76\\74&45\end{bmatrix}$, $\begin{bmatrix}61&30\\28&41\end{bmatrix}$, $\begin{bmatrix}69&46\\46&27\end{bmatrix}$, $\begin{bmatrix}75&16\\22&15\end{bmatrix}$, $\begin{bmatrix}75&52\\70&9\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	84.96.1-84.b.1.1, 84.96.1-84.b.1.2, 84.96.1-84.b.1.3, 84.96.1-84.b.1.4, 84.96.1-84.b.1.5, 84.96.1-84.b.1.6, 84.96.1-84.b.1.7, 84.96.1-84.b.1.8, 84.96.1-84.b.1.9, 84.96.1-84.b.1.10, 84.96.1-84.b.1.11, 84.96.1-84.b.1.12, 84.96.1-84.b.1.13, 84.96.1-84.b.1.14, 84.96.1-84.b.1.15, 84.96.1-84.b.1.16, 84.96.1-84.b.1.17, 84.96.1-84.b.1.18, 84.96.1-84.b.1.19, 84.96.1-84.b.1.20, 168.96.1-84.b.1.1, 168.96.1-84.b.1.2, 168.96.1-84.b.1.3, 168.96.1-84.b.1.4, 168.96.1-84.b.1.5, 168.96.1-84.b.1.6, 168.96.1-84.b.1.7, 168.96.1-84.b.1.8, 168.96.1-84.b.1.9, 168.96.1-84.b.1.10, 168.96.1-84.b.1.11, 168.96.1-84.b.1.12, 168.96.1-84.b.1.13, 168.96.1-84.b.1.14, 168.96.1-84.b.1.15, 168.96.1-84.b.1.16, 168.96.1-84.b.1.17, 168.96.1-84.b.1.18, 168.96.1-84.b.1.19, 168.96.1-84.b.1.20
Cyclic 84-isogeny field degree:	$16$
Cyclic 84-torsion field degree:	$384$
Full 84-torsion field degree:	$193536$

Jacobian

Conductor:	$?$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	not computed

Rational points

This modular curve is an elliptic curve, but the rank has not been computed

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_0(3)$	$3$	$12$	$12$	$0$	$0$	full Jacobian
28.12.0.b.1	$28$	$4$	$4$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\pm1}(2,6)$	$6$	$2$	$2$	$0$	$0$	full Jacobian
28.12.0.b.1	$28$	$4$	$4$	$0$	$0$	full Jacobian
84.24.0.o.1	$84$	$2$	$2$	$0$	$?$	full Jacobian
84.24.1.o.1	$84$	$2$	$2$	$1$	$?$	dimension zero

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
84.96.1.f.1	$84$	$2$	$2$	$1$	$?$	dimension zero
84.96.1.f.2	$84$	$2$	$2$	$1$	$?$	dimension zero
84.96.1.f.3	$84$	$2$	$2$	$1$	$?$	dimension zero
84.96.1.f.4	$84$	$2$	$2$	$1$	$?$	dimension zero
84.96.3.b.1	$84$	$2$	$2$	$3$	$?$	not computed
84.96.3.c.1	$84$	$2$	$2$	$3$	$?$	not computed
84.96.3.h.1	$84$	$2$	$2$	$3$	$?$	not computed
84.96.3.j.1	$84$	$2$	$2$	$3$	$?$	not computed
84.96.3.p.1	$84$	$2$	$2$	$3$	$?$	not computed
84.96.3.p.2	$84$	$2$	$2$	$3$	$?$	not computed
84.96.3.t.1	$84$	$2$	$2$	$3$	$?$	not computed
84.96.3.t.2	$84$	$2$	$2$	$3$	$?$	not computed
84.144.5.b.1	$84$	$3$	$3$	$5$	$?$	not computed
168.96.1.lq.1	$168$	$2$	$2$	$1$	$?$	dimension zero
168.96.1.lq.2	$168$	$2$	$2$	$1$	$?$	dimension zero
168.96.1.lq.3	$168$	$2$	$2$	$1$	$?$	dimension zero
168.96.1.lq.4	$168$	$2$	$2$	$1$	$?$	dimension zero
168.96.3.ct.1	$168$	$2$	$2$	$3$	$?$	not computed
168.96.3.cw.1	$168$	$2$	$2$	$3$	$?$	not computed
168.96.3.di.1	$168$	$2$	$2$	$3$	$?$	not computed
168.96.3.do.1	$168$	$2$	$2$	$3$	$?$	not computed
168.96.3.eq.1	$168$	$2$	$2$	$3$	$?$	not computed
168.96.3.eq.2	$168$	$2$	$2$	$3$	$?$	not computed
168.96.3.fj.1	$168$	$2$	$2$	$3$	$?$	not computed
168.96.3.fj.2	$168$	$2$	$2$	$3$	$?$	not computed
252.144.5.b.1	$252$	$3$	$3$	$5$	$?$	not computed
252.144.9.b.1	$252$	$3$	$3$	$9$	$?$	not computed
252.144.9.f.1	$252$	$3$	$3$	$9$	$?$	not computed