Invariants
Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12G3 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}19&46\\40&77\end{bmatrix}$, $\begin{bmatrix}21&47\\52&69\end{bmatrix}$, $\begin{bmatrix}41&43\\16&7\end{bmatrix}$, $\begin{bmatrix}71&59\\32&81\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 84.144.3-84.gm.1.1, 84.144.3-84.gm.1.2, 84.144.3-84.gm.1.3, 84.144.3-84.gm.1.4, 168.144.3-84.gm.1.1, 168.144.3-84.gm.1.2, 168.144.3-84.gm.1.3, 168.144.3-84.gm.1.4, 168.144.3-84.gm.1.5, 168.144.3-84.gm.1.6, 168.144.3-84.gm.1.7, 168.144.3-84.gm.1.8, 168.144.3-84.gm.1.9, 168.144.3-84.gm.1.10, 168.144.3-84.gm.1.11, 168.144.3-84.gm.1.12 |
Cyclic 84-isogeny field degree: | $32$ |
Cyclic 84-torsion field degree: | $768$ |
Full 84-torsion field degree: | $129024$ |
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 |
---|---|---|---|---|---|
12.36.2.ba.1 | $12$ | $2$ | $2$ | $2$ | $0$ |
84.36.0.bg.1 | $84$ | $2$ | $2$ | $0$ | $?$ |
84.36.1.j.1 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.36.1.cw.1 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.36.1.dw.1 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.36.2.s.1 | $84$ | $2$ | $2$ | $2$ | $?$ |
84.36.2.cm.1 | $84$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.144.7.emk.1 | $168$ | $2$ | $2$ | $7$ |
168.144.7.eml.1 | $168$ | $2$ | $2$ | $7$ |
168.144.7.enq.1 | $168$ | $2$ | $2$ | $7$ |
168.144.7.enr.1 | $168$ | $2$ | $2$ | $7$ |
168.144.7.eow.1 | $168$ | $2$ | $2$ | $7$ |
168.144.7.eox.1 | $168$ | $2$ | $2$ | $7$ |
168.144.7.epe.1 | $168$ | $2$ | $2$ | $7$ |
168.144.7.epf.1 | $168$ | $2$ | $2$ | $7$ |
252.216.15.iv.1 | $252$ | $3$ | $3$ | $15$ |