Invariants
| Level: | $84$ | $\SL_2$-level: | $6$ | Newform level: | $144$ | ||
| 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 | $6^{12}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| 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: | 6F1 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}18&31\\79&48\end{bmatrix}$, $\begin{bmatrix}31&60\\78&67\end{bmatrix}$, $\begin{bmatrix}44&9\\45&44\end{bmatrix}$, $\begin{bmatrix}60&71\\35&66\end{bmatrix}$, $\begin{bmatrix}69&32\\62&69\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.72.1.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: | $64512$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 1 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(y^{2}-3z^{2})^{3}(y^{6}-225y^{4}z^{2}-405y^{2}z^{4}-243z^{6})^{3}}{z^{2}y^{6}(y^{2}+z^{2})^{2}(y^{2}+9z^{2})^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| 4.6.0.a.1 | $4$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 21.24.0-3.a.1.1 | $21$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
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.d.1.3 | $84$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 84.48.0-12.d.1.6 | $84$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 84.48.1-12.b.1.1 | $84$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 84.72.0-6.a.1.4 | $84$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 84.288.3-12.b.1.3 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.288.3-12.b.1.4 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.288.3-84.b.1.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.288.3-84.b.1.13 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.288.5-12.i.1.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-12.k.1.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-12.l.1.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-12.n.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.ba.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.bc.1.4 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.be.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.5-84.bg.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.288.7-12.bb.1.7 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-12.bb.1.8 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-84.el.1.6 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.288.7-84.el.1.10 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.3-24.b.1.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.3-24.b.1.8 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.3-168.b.1.9 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.3-168.b.1.21 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.5-24.cf.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.cs.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.cz.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.dn.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.ha.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.ho.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.ic.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.iq.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.7-24.or.1.15 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-24.or.1.16 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.bxh.1.14 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.bxh.1.18 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 252.432.7-36.c.1.4 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-36.c.1.6 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.c.1.9 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.c.1.17 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-36.d.1.4 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.d.1.9 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.d.1.17 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-36.e.1.4 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.e.1.9 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.7-252.e.1.17 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 252.432.10-36.a.1.1 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.10-36.a.1.3 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 252.432.13-36.b.1.2 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |