Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2\cdot4\cdot6\cdot12$ | Cusp orbits | $1^{4}$ | ||
| 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: | 12F1 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}16&15\\45&10\end{bmatrix}$, $\begin{bmatrix}28&59\\39&74\end{bmatrix}$, $\begin{bmatrix}72&61\\49&48\end{bmatrix}$, $\begin{bmatrix}82&35\\81&32\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.24.1.o.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $384$ |
| Full 84-torsion field degree: | $193536$ |
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 | $3$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 28.6.0.d.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 |
|---|---|---|---|---|---|---|
| 6.24.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 84.24.0-6.a.1.9 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 84.96.1-84.b.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.e.1.12 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.j.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.k.1.12 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.y.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.ba.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.bc.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.be.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.144.3-84.ma.1.1 | $84$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 84.384.13-84.z.1.32 | $84$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 168.96.1-168.gl.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.jy.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.yz.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.zc.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bkw.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.blc.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bli.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.blo.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 252.144.3-252.ce.1.2 | $252$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 252.144.5-252.p.1.15 | $252$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 252.144.5-252.t.1.2 | $252$ | $3$ | $3$ | $5$ | $?$ | not computed |