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Properties

Label 84.192.11.bm.1
Level $84$
Index $192$
Genus $11$
Cusps $12$
$\Q$-cusps $12$

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Invariants

Level: $84$ $\SL_2$-level: $84$ Newform level: $1$
Index: $192$ $\PSL_2$-index:$192$
Genus: $11 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (all of which are rational) Cusp widths $1^{2}\cdot3^{2}\cdot4\cdot7^{2}\cdot12\cdot21^{2}\cdot28\cdot84$ Cusp orbits $1^{12}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $4 \le \gamma \le 11$
$\overline{\Q}$-gonality: $4 \le \gamma \le 11$
Rational cusps: $12$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 84I11

Level structure

$\GL_2(\Z/84\Z)$-generators: $\begin{bmatrix}5&34\\0&47\end{bmatrix}$, $\begin{bmatrix}31&70\\0&73\end{bmatrix}$, $\begin{bmatrix}43&3\\0&53\end{bmatrix}$, $\begin{bmatrix}53&22\\0&19\end{bmatrix}$, $\begin{bmatrix}53&62\\0&37\end{bmatrix}$, $\begin{bmatrix}59&77\\0&29\end{bmatrix}$, $\begin{bmatrix}65&4\\0&67\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 84.384.11-84.bm.1.1, 84.384.11-84.bm.1.2, 84.384.11-84.bm.1.3, 84.384.11-84.bm.1.4, 84.384.11-84.bm.1.5, 84.384.11-84.bm.1.6, 84.384.11-84.bm.1.7, 84.384.11-84.bm.1.8, 84.384.11-84.bm.1.9, 84.384.11-84.bm.1.10, 84.384.11-84.bm.1.11, 84.384.11-84.bm.1.12, 84.384.11-84.bm.1.13, 84.384.11-84.bm.1.14, 84.384.11-84.bm.1.15, 84.384.11-84.bm.1.16, 84.384.11-84.bm.1.17, 84.384.11-84.bm.1.18, 84.384.11-84.bm.1.19, 84.384.11-84.bm.1.20, 84.384.11-84.bm.1.21, 84.384.11-84.bm.1.22, 84.384.11-84.bm.1.23, 84.384.11-84.bm.1.24, 84.384.11-84.bm.1.25, 84.384.11-84.bm.1.26, 84.384.11-84.bm.1.27, 84.384.11-84.bm.1.28, 84.384.11-84.bm.1.29, 84.384.11-84.bm.1.30, 84.384.11-84.bm.1.31, 84.384.11-84.bm.1.32, 84.384.11-84.bm.1.33, 84.384.11-84.bm.1.34, 84.384.11-84.bm.1.35, 84.384.11-84.bm.1.36, 84.384.11-84.bm.1.37, 84.384.11-84.bm.1.38, 84.384.11-84.bm.1.39, 84.384.11-84.bm.1.40, 84.384.11-84.bm.1.41, 84.384.11-84.bm.1.42, 84.384.11-84.bm.1.43, 84.384.11-84.bm.1.44, 84.384.11-84.bm.1.45, 84.384.11-84.bm.1.46, 84.384.11-84.bm.1.47, 84.384.11-84.bm.1.48, 168.384.11-84.bm.1.1, 168.384.11-84.bm.1.2, 168.384.11-84.bm.1.3, 168.384.11-84.bm.1.4, 168.384.11-84.bm.1.5, 168.384.11-84.bm.1.6, 168.384.11-84.bm.1.7, 168.384.11-84.bm.1.8, 168.384.11-84.bm.1.9, 168.384.11-84.bm.1.10, 168.384.11-84.bm.1.11, 168.384.11-84.bm.1.12, 168.384.11-84.bm.1.13, 168.384.11-84.bm.1.14, 168.384.11-84.bm.1.15, 168.384.11-84.bm.1.16, 168.384.11-84.bm.1.17, 168.384.11-84.bm.1.18, 168.384.11-84.bm.1.19, 168.384.11-84.bm.1.20, 168.384.11-84.bm.1.21, 168.384.11-84.bm.1.22, 168.384.11-84.bm.1.23, 168.384.11-84.bm.1.24, 168.384.11-84.bm.1.25, 168.384.11-84.bm.1.26, 168.384.11-84.bm.1.27, 168.384.11-84.bm.1.28, 168.384.11-84.bm.1.29, 168.384.11-84.bm.1.30, 168.384.11-84.bm.1.31, 168.384.11-84.bm.1.32, 168.384.11-84.bm.1.33, 168.384.11-84.bm.1.34, 168.384.11-84.bm.1.35, 168.384.11-84.bm.1.36, 168.384.11-84.bm.1.37, 168.384.11-84.bm.1.38, 168.384.11-84.bm.1.39, 168.384.11-84.bm.1.40, 168.384.11-84.bm.1.41, 168.384.11-84.bm.1.42, 168.384.11-84.bm.1.43, 168.384.11-84.bm.1.44, 168.384.11-84.bm.1.45, 168.384.11-84.bm.1.46, 168.384.11-84.bm.1.47, 168.384.11-84.bm.1.48, 168.384.11-84.bm.1.49, 168.384.11-84.bm.1.50, 168.384.11-84.bm.1.51, 168.384.11-84.bm.1.52, 168.384.11-84.bm.1.53, 168.384.11-84.bm.1.54, 168.384.11-84.bm.1.55, 168.384.11-84.bm.1.56, 168.384.11-84.bm.1.57, 168.384.11-84.bm.1.58, 168.384.11-84.bm.1.59, 168.384.11-84.bm.1.60, 168.384.11-84.bm.1.61, 168.384.11-84.bm.1.62, 168.384.11-84.bm.1.63, 168.384.11-84.bm.1.64, 168.384.11-84.bm.1.65, 168.384.11-84.bm.1.66, 168.384.11-84.bm.1.67, 168.384.11-84.bm.1.68, 168.384.11-84.bm.1.69, 168.384.11-84.bm.1.70, 168.384.11-84.bm.1.71, 168.384.11-84.bm.1.72, 168.384.11-84.bm.1.73, 168.384.11-84.bm.1.74, 168.384.11-84.bm.1.75, 168.384.11-84.bm.1.76, 168.384.11-84.bm.1.77, 168.384.11-84.bm.1.78, 168.384.11-84.bm.1.79, 168.384.11-84.bm.1.80, 168.384.11-84.bm.1.81, 168.384.11-84.bm.1.82, 168.384.11-84.bm.1.83, 168.384.11-84.bm.1.84, 168.384.11-84.bm.1.85, 168.384.11-84.bm.1.86, 168.384.11-84.bm.1.87, 168.384.11-84.bm.1.88, 168.384.11-84.bm.1.89, 168.384.11-84.bm.1.90, 168.384.11-84.bm.1.91, 168.384.11-84.bm.1.92, 168.384.11-84.bm.1.93, 168.384.11-84.bm.1.94, 168.384.11-84.bm.1.95, 168.384.11-84.bm.1.96, 168.384.11-84.bm.1.97, 168.384.11-84.bm.1.98, 168.384.11-84.bm.1.99, 168.384.11-84.bm.1.100, 168.384.11-84.bm.1.101, 168.384.11-84.bm.1.102, 168.384.11-84.bm.1.103, 168.384.11-84.bm.1.104, 168.384.11-84.bm.1.105, 168.384.11-84.bm.1.106, 168.384.11-84.bm.1.107, 168.384.11-84.bm.1.108, 168.384.11-84.bm.1.109, 168.384.11-84.bm.1.110, 168.384.11-84.bm.1.111, 168.384.11-84.bm.1.112
Cyclic 84-isogeny field degree: $1$
Cyclic 84-torsion field degree: $24$
Full 84-torsion field degree: $48384$

Rational points

This modular curve has 12 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
$X_0(3)$ $3$ $48$ $48$ $0$ $0$
$X_0(4)$ $4$ $32$ $32$ $0$ $0$
$X_0(7)$ $7$ $24$ $24$ $0$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X_0(12)$ $12$ $8$ $8$ $0$ $0$
$X_0(28)$ $28$ $4$ $4$ $2$ $0$
$X_0(42)$ $42$ $2$ $2$ $5$ $0$

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

Covering curve Level Index Degree Genus
84.384.21.bk.1 $84$ $2$ $2$ $21$
84.384.21.bk.2 $84$ $2$ $2$ $21$
84.384.21.bk.3 $84$ $2$ $2$ $21$
84.384.21.bk.4 $84$ $2$ $2$ $21$
84.384.21.bl.1 $84$ $2$ $2$ $21$
84.384.21.bl.2 $84$ $2$ $2$ $21$
84.384.21.bl.3 $84$ $2$ $2$ $21$
84.384.21.bl.4 $84$ $2$ $2$ $21$
84.384.23.i.1 $84$ $2$ $2$ $23$
84.384.23.i.2 $84$ $2$ $2$ $23$
84.384.23.i.3 $84$ $2$ $2$ $23$
84.384.23.i.4 $84$ $2$ $2$ $23$
84.384.23.j.1 $84$ $2$ $2$ $23$
84.384.23.j.2 $84$ $2$ $2$ $23$
84.384.23.j.3 $84$ $2$ $2$ $23$
84.384.23.j.4 $84$ $2$ $2$ $23$
84.384.23.k.1 $84$ $2$ $2$ $23$
84.384.23.k.2 $84$ $2$ $2$ $23$
84.384.23.k.3 $84$ $2$ $2$ $23$
84.384.23.k.4 $84$ $2$ $2$ $23$
84.384.23.l.1 $84$ $2$ $2$ $23$
84.384.23.l.2 $84$ $2$ $2$ $23$
84.384.23.l.3 $84$ $2$ $2$ $23$
84.384.23.l.4 $84$ $2$ $2$ $23$
168.384.21.bgx.1 $168$ $2$ $2$ $21$
168.384.21.bgx.2 $168$ $2$ $2$ $21$
168.384.21.bgx.3 $168$ $2$ $2$ $21$
168.384.21.bgx.4 $168$ $2$ $2$ $21$
168.384.21.bha.1 $168$ $2$ $2$ $21$
168.384.21.bha.2 $168$ $2$ $2$ $21$
168.384.21.bha.3 $168$ $2$ $2$ $21$
168.384.21.bha.4 $168$ $2$ $2$ $21$
168.384.23.ls.1 $168$ $2$ $2$ $23$
168.384.23.ls.2 $168$ $2$ $2$ $23$
168.384.23.lt.1 $168$ $2$ $2$ $23$
168.384.23.lt.2 $168$ $2$ $2$ $23$
168.384.23.lw.1 $168$ $2$ $2$ $23$
168.384.23.lw.2 $168$ $2$ $2$ $23$
168.384.23.lx.1 $168$ $2$ $2$ $23$
168.384.23.lx.2 $168$ $2$ $2$ $23$
168.384.23.ma.1 $168$ $2$ $2$ $23$
168.384.23.ma.2 $168$ $2$ $2$ $23$
168.384.23.mb.1 $168$ $2$ $2$ $23$
168.384.23.mb.2 $168$ $2$ $2$ $23$
168.384.23.me.1 $168$ $2$ $2$ $23$
168.384.23.me.2 $168$ $2$ $2$ $23$
168.384.23.mf.1 $168$ $2$ $2$ $23$
168.384.23.mf.2 $168$ $2$ $2$ $23$
168.384.23.mg.1 $168$ $2$ $2$ $23$
168.384.23.mg.2 $168$ $2$ $2$ $23$
168.384.23.mg.3 $168$ $2$ $2$ $23$
168.384.23.mg.4 $168$ $2$ $2$ $23$
168.384.23.mh.1 $168$ $2$ $2$ $23$
168.384.23.mh.2 $168$ $2$ $2$ $23$
168.384.23.mh.3 $168$ $2$ $2$ $23$
168.384.23.mh.4 $168$ $2$ $2$ $23$
168.384.23.mi.1 $168$ $2$ $2$ $23$
168.384.23.mi.2 $168$ $2$ $2$ $23$
168.384.23.mi.3 $168$ $2$ $2$ $23$
168.384.23.mi.4 $168$ $2$ $2$ $23$
168.384.23.mj.1 $168$ $2$ $2$ $23$
168.384.23.mj.2 $168$ $2$ $2$ $23$
168.384.23.mj.3 $168$ $2$ $2$ $23$
168.384.23.mj.4 $168$ $2$ $2$ $23$