Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $36$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $3^{8}\cdot12^{4}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
| 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: | 12S1 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}7&74\\45&59\end{bmatrix}$, $\begin{bmatrix}53&60\\9&71\end{bmatrix}$, $\begin{bmatrix}79&60\\57&73\end{bmatrix}$, $\begin{bmatrix}83&46\\51&73\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.72.1.h.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $384$ |
| Full 84-torsion field degree: | $64512$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 27 $ |
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 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^6}\cdot\frac{(y^{2}+81z^{2})^{3}(y^{6}+243y^{4}z^{2}+177147y^{2}z^{4}+4782969z^{6})^{3}}{z^{4}y^{12}(y^{2}+27z^{2})(y^{2}+243z^{2})^{3}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.72.0-6.a.1.1 | $42$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 84.48.0-12.j.1.6 | $84$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 84.48.0-12.j.1.8 | $84$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 84.72.0-6.a.1.3 | $84$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 84.288.5-12.e.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-12.k.1.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-12.z.1.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-12.bc.1.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.ea.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.eb.1.3 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.fo.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.fp.1.3 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.bk.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.cw.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.hz.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.iu.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bfv.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bgc.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bqo.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bqv.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 252.432.7-36.z.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-36.z.1.4 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-36.bf.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-36.bm.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.da.1.2 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.da.1.11 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.dj.1.3 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.dj.1.10 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.eg.1.3 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.eg.1.10 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.10-36.y.1.1 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-36.y.1.7 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.13-36.z.1.1 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |