Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $72$ | ||
| 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: | $\begin{bmatrix}2&55\\57&64\end{bmatrix}$, $\begin{bmatrix}17&0\\48&29\end{bmatrix}$, $\begin{bmatrix}44&49\\59&60\end{bmatrix}$, $\begin{bmatrix}46&71\\43&36\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.1.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: | $193536$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 72.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 219x + 1190 $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^6}\cdot\frac{12x^{2}y^{6}-24543x^{2}y^{4}z^{2}+13401936x^{2}y^{2}z^{4}-2423430009x^{2}z^{6}-294xy^{6}z+417960xy^{4}z^{3}-228836745xy^{2}z^{5}+41794764102xz^{7}-y^{8}+2496y^{6}z^{2}-2288088y^{4}z^{4}+1024189596y^{2}z^{6}-175992060609z^{8}}{z^{4}y^{2}(24x^{2}-408xz-y^{2}+1680z^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.24.0-6.a.1.4 | $42$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 84.24.0-6.a.1.5 | $84$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 84.96.1-12.d.1.10 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-12.f.1.6 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-12.j.1.4 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-12.k.1.6 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.bg.1.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.bh.1.3 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.bk.1.6 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.bl.1.5 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.144.3-12.cf.1.1 | $84$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 84.384.13-84.bb.1.28 | $84$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 168.96.1-24.cr.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.dx.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.ij.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.im.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byk.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byn.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byw.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byz.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 252.144.3-36.u.1.7 | $252$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 252.144.5-36.h.1.6 | $252$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 252.144.5-36.l.1.8 | $252$ | $3$ | $3$ | $5$ | $?$ | not computed |