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: | $\begin{bmatrix}1&15\\58&77\end{bmatrix}$, $\begin{bmatrix}19&68\\60&59\end{bmatrix}$, $\begin{bmatrix}65&76\\38&3\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 84.48.1.i.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 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 | Kernel decomposition |
---|---|---|---|---|---|---|
12.48.0-12.f.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
42.48.0-42.b.1.2 | $42$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
84.48.0-42.b.1.2 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
84.48.0-12.f.1.6 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
84.48.1-84.p.1.7 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
84.48.1-84.p.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.bw.1.3 | $84$ | $3$ | $3$ | $5$ | $?$ | not computed |
168.192.3-168.ki.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.ki.2.3 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.kk.1.5 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.kk.2.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.oa.1.8 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.oa.2.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.oc.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.oc.2.3 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
252.288.5-252.i.1.8 | $252$ | $3$ | $3$ | $5$ | $?$ | not computed |
252.288.9-252.q.1.5 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
252.288.9-252.y.1.6 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |