Invariants
Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12P1 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}41&63\\42&41\end{bmatrix}$, $\begin{bmatrix}73&33\\70&17\end{bmatrix}$, $\begin{bmatrix}77&58\\50&63\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 84.48.1.j.1 for the level structure with $-I$) |
Cyclic 84-isogeny field degree: | $16$ |
Cyclic 84-torsion field degree: | $384$ |
Full 84-torsion field degree: | $96768$ |
Jacobian
Conductor: | $?$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | not computed |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
12.48.0-12.f.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
84.48.0-12.f.1.5 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
84.48.0-84.m.1.6 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
84.48.0-84.m.1.7 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
84.48.1-84.o.1.7 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
84.48.1-84.o.1.16 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
84.288.5-84.bx.1.3 | $84$ | $3$ | $3$ | $5$ | $?$ | not computed |
168.192.3-168.km.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.km.2.13 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.ko.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.ko.2.13 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.oe.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.oe.2.13 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.og.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.og.2.13 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
252.288.5-252.j.1.8 | $252$ | $3$ | $3$ | $5$ | $?$ | not computed |
252.288.9-252.r.1.2 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
252.288.9-252.z.1.7 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |