Invariants
Level: | $84$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4E0 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}7&82\\66&59\end{bmatrix}$, $\begin{bmatrix}23&30\\36&55\end{bmatrix}$, $\begin{bmatrix}57&34\\28&39\end{bmatrix}$, $\begin{bmatrix}59&22\\26&41\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.12.0.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: | $387072$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 621 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^2}\cdot\frac{x^{12}(9x^{4}+192x^{2}y^{2}+4096y^{4})^{3}}{y^{4}x^{16}(3x^{2}+64y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
28.12.0-2.a.1.1 | $28$ | $2$ | $2$ | $0$ | $0$ |
84.12.0-2.a.1.1 | $84$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
84.48.0-12.b.1.1 | $84$ | $2$ | $2$ | $0$ |
84.48.0-12.b.1.2 | $84$ | $2$ | $2$ | $0$ |
84.48.0-12.c.1.1 | $84$ | $2$ | $2$ | $0$ |
84.48.0-12.c.1.4 | $84$ | $2$ | $2$ | $0$ |
84.48.0-84.f.1.2 | $84$ | $2$ | $2$ | $0$ |
84.48.0-84.f.1.7 | $84$ | $2$ | $2$ | $0$ |
84.48.0-84.g.1.2 | $84$ | $2$ | $2$ | $0$ |
84.48.0-84.g.1.8 | $84$ | $2$ | $2$ | $0$ |
84.72.2-12.d.1.3 | $84$ | $3$ | $3$ | $2$ |
84.96.1-12.d.1.5 | $84$ | $4$ | $4$ | $1$ |
84.192.5-84.b.1.1 | $84$ | $8$ | $8$ | $5$ |
84.504.16-84.b.1.3 | $84$ | $21$ | $21$ | $16$ |
168.48.0-24.d.1.3 | $168$ | $2$ | $2$ | $0$ |
168.48.0-24.d.1.5 | $168$ | $2$ | $2$ | $0$ |
168.48.0-24.g.1.3 | $168$ | $2$ | $2$ | $0$ |
168.48.0-24.g.1.5 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.n.1.7 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.n.1.10 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.q.1.7 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.q.1.10 | $168$ | $2$ | $2$ | $0$ |