Invariants
Level: | $84$ | $\SL_2$-level: | $4$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4G0 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}11&76\\14&9\end{bmatrix}$, $\begin{bmatrix}51&16\\20&11\end{bmatrix}$, $\begin{bmatrix}53&34\\22&63\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.24.0.b.1 for the level structure with $-I$) |
Cyclic 84-isogeny field degree: | $64$ |
Cyclic 84-torsion field degree: | $1536$ |
Full 84-torsion field degree: | $193536$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 12 x^{2} + y^{2} + y z + z^{2} $ |
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 |
---|---|---|---|---|---|
28.24.0-4.a.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ |
84.24.0-4.a.1.1 | $84$ | $2$ | $2$ | $0$ | $?$ |
84.24.0-12.b.1.1 | $84$ | $2$ | $2$ | $0$ | $?$ |
84.24.0-12.b.1.2 | $84$ | $2$ | $2$ | $0$ | $?$ |
84.24.0-12.b.1.4 | $84$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
84.144.4-12.d.1.3 | $84$ | $3$ | $3$ | $4$ |
84.192.3-12.d.1.2 | $84$ | $4$ | $4$ | $3$ |
168.96.1-24.j.1.4 | $168$ | $2$ | $2$ | $1$ |
168.96.1-24.k.1.4 | $168$ | $2$ | $2$ | $1$ |
168.96.1-24.bp.1.4 | $168$ | $2$ | $2$ | $1$ |
168.96.1-24.bq.1.4 | $168$ | $2$ | $2$ | $1$ |
84.384.11-84.b.1.7 | $84$ | $8$ | $8$ | $11$ |
168.96.1-168.bo.1.3 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.bp.1.7 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.cu.1.3 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.cv.1.7 | $168$ | $2$ | $2$ | $1$ |