Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12V1 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}43&60\\58&65\end{bmatrix}$, $\begin{bmatrix}55&0\\26&73\end{bmatrix}$, $\begin{bmatrix}55&6\\26&55\end{bmatrix}$, $\begin{bmatrix}67&72\\32&73\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.96.1.f.3 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $192$ |
| Full 84-torsion field degree: | $48384$ |
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 | $3$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
| 28.24.0-28.b.1.2 | $28$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.0-12.a.2.15 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 84.96.0-12.a.2.11 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 84.96.0-84.a.2.8 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 84.96.0-84.a.2.23 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 84.96.1-84.b.1.2 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.b.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 84.384.5-84.g.1.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.h.4.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.m.3.3 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.o.3.7 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.q.2.6 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.r.2.6 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.w.2.5 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.y.3.6 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.iu.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ja.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ke.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ks.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.og.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.om.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.pq.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.qe.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |