Invariants
Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (none of which are rational) | Cusp widths | $6^{8}\cdot12^{8}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
$\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12B5 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}23&4\\0&19\end{bmatrix}$, $\begin{bmatrix}35&6\\9&35\end{bmatrix}$, $\begin{bmatrix}41&16\\27&43\end{bmatrix}$, $\begin{bmatrix}47&22\\42&55\end{bmatrix}$, $\begin{bmatrix}53&36\\72&29\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 84.288.5-84.be.1.1, 84.288.5-84.be.1.2, 84.288.5-84.be.1.3, 84.288.5-84.be.1.4, 84.288.5-84.be.1.5, 84.288.5-84.be.1.6, 84.288.5-84.be.1.7, 84.288.5-84.be.1.8, 168.288.5-84.be.1.1, 168.288.5-84.be.1.2, 168.288.5-84.be.1.3, 168.288.5-84.be.1.4, 168.288.5-84.be.1.5, 168.288.5-84.be.1.6, 168.288.5-84.be.1.7, 168.288.5-84.be.1.8 |
Cyclic 84-isogeny field degree: | $16$ |
Cyclic 84-torsion field degree: | $384$ |
Full 84-torsion field degree: | $64512$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.72.1.b.1 | $12$ | $2$ | $2$ | $1$ | $0$ |
84.48.1.f.1 | $84$ | $3$ | $3$ | $1$ | $?$ |
84.72.1.r.1 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.72.1.cb.1 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.72.3.cn.1 | $84$ | $2$ | $2$ | $3$ | $?$ |
84.72.3.dd.1 | $84$ | $2$ | $2$ | $3$ | $?$ |
84.72.3.jx.1 | $84$ | $2$ | $2$ | $3$ | $?$ |
84.72.3.qz.1 | $84$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
84.288.13.da.1 | $84$ | $2$ | $2$ | $13$ |
84.288.13.db.1 | $84$ | $2$ | $2$ | $13$ |
168.288.13.bci.1 | $168$ | $2$ | $2$ | $13$ |
168.288.13.bcj.1 | $168$ | $2$ | $2$ | $13$ |
252.432.21.et.1 | $252$ | $3$ | $3$ | $21$ |
252.432.21.ez.1 | $252$ | $3$ | $3$ | $21$ |
252.432.21.fo.1 | $252$ | $3$ | $3$ | $21$ |
252.432.21.ft.1 | $252$ | $3$ | $3$ | $21$ |
252.432.21.fz.1 | $252$ | $3$ | $3$ | $21$ |
252.432.21.ge.1 | $252$ | $3$ | $3$ | $21$ |