Invariants
Level: | $84$ | $\SL_2$-level: | $12$ | 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 | $3^{8}\cdot12^{4}$ | 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: | 12S1 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}43&6\\66&61\end{bmatrix}$, $\begin{bmatrix}55&68\\42&17\end{bmatrix}$, $\begin{bmatrix}61&44\\81&71\end{bmatrix}$, $\begin{bmatrix}71&36\\51&35\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.72.1.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: | $64512$ |
Jacobian
Conductor: | $?$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 144.2.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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^6}\cdot\frac{(y-9z)^{3}(y+9z)^{3}(y^{3}-27y^{2}z+243yz^{2}+2187z^{3})^{3}(y^{3}+27y^{2}z+243yz^{2}-2187z^{3})^{3}}{z^{4}y^{12}(y^{2}-243z^{2})^{3}(y^{2}-27z^{2})}$ |
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.i.1.6 | $84$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
84.48.0-12.i.1.8 | $84$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
84.72.0-6.a.1.2 | $84$ | $2$ | $2$ | $0$ | $?$ | 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.f.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-12.l.1.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-12.y.1.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-12.bd.1.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-84.ec.1.3 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-84.ee.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-84.fq.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-84.fs.1.4 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.bl.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.dd.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.hs.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.jb.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.bgj.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.bgx.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.brc.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.brq.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
252.432.7-36.ba.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-36.ba.1.3 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-36.bg.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-36.bo.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.db.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.db.1.10 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.dk.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.dk.1.10 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.eh.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.eh.1.11 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.10-36.z.1.1 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
252.432.10-36.z.1.5 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
252.432.13-36.ba.1.1 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |