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: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}5&36\\68&25\end{bmatrix}$, $\begin{bmatrix}27&74\\46&23\end{bmatrix}$, $\begin{bmatrix}29&82\\74&81\end{bmatrix}$, $\begin{bmatrix}53&12\\62&73\end{bmatrix}$, $\begin{bmatrix}63&82\\34&69\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.48.1.b.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 is an elliptic curve, but the rank has not been computed
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 28.12.0.b.1 | $28$ | $8$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.48.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 84.48.0-6.a.1.7 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 84.48.0-84.o.1.2 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 84.48.0-84.o.1.15 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 84.48.1-84.o.1.1 | $84$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 84.192.1-84.f.1.2 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.192.1-84.f.2.2 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.192.1-84.f.3.4 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.192.1-84.f.4.4 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.192.3-84.b.1.9 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.c.1.12 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.h.1.4 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.j.1.12 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.p.1.3 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.p.2.8 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.t.1.4 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.t.2.8 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.288.5-84.b.1.2 | $84$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.192.1-168.lq.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lq.2.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lq.3.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lq.4.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.3-168.ct.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.cw.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.di.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.do.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.eq.1.10 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.eq.2.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.fj.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.fj.2.10 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 252.288.5-252.b.1.2 | $252$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 252.288.9-252.b.1.13 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 252.288.9-252.f.1.16 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |