Invariants
Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
Cusps: | $24$ (none of which are rational) | Cusp widths | $4^{12}\cdot12^{12}$ | Cusp orbits | $2^{4}\cdot4^{4}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12E5 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}1&60\\34&47\end{bmatrix}$, $\begin{bmatrix}3&58\\32&79\end{bmatrix}$, $\begin{bmatrix}41&66\\12&59\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 84.192.5.o.3 for the level structure with $-I$) |
Cyclic 84-isogeny field degree: | $16$ |
Cyclic 84-torsion field degree: | $192$ |
Full 84-torsion field degree: | $24192$ |
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 |
---|---|---|---|---|---|
12.192.1-12.d.1.8 | $12$ | $2$ | $2$ | $1$ | $0$ |
84.192.1-12.d.1.2 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.192.1-84.f.3.2 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.192.1-84.f.3.16 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.192.1-84.g.4.2 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.192.1-84.g.4.14 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.192.3-84.j.1.2 | $84$ | $2$ | $2$ | $3$ | $?$ |
84.192.3-84.j.1.7 | $84$ | $2$ | $2$ | $3$ | $?$ |
84.192.3-84.n.1.4 | $84$ | $2$ | $2$ | $3$ | $?$ |
84.192.3-84.n.1.5 | $84$ | $2$ | $2$ | $3$ | $?$ |
84.192.3-84.p.2.4 | $84$ | $2$ | $2$ | $3$ | $?$ |
84.192.3-84.p.2.11 | $84$ | $2$ | $2$ | $3$ | $?$ |
84.192.3-84.q.1.6 | $84$ | $2$ | $2$ | $3$ | $?$ |
84.192.3-84.q.1.15 | $84$ | $2$ | $2$ | $3$ | $?$ |