Invariants
Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $72$ | $\PSL_2$-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: | $\begin{bmatrix}13&24\\3&79\end{bmatrix}$, $\begin{bmatrix}53&82\\24&19\end{bmatrix}$, $\begin{bmatrix}59&24\\33&59\end{bmatrix}$, $\begin{bmatrix}67&74\\57&65\end{bmatrix}$, $\begin{bmatrix}73&68\\27&71\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 84.144.1-84.r.1.1, 84.144.1-84.r.1.2, 84.144.1-84.r.1.3, 84.144.1-84.r.1.4, 84.144.1-84.r.1.5, 84.144.1-84.r.1.6, 84.144.1-84.r.1.7, 84.144.1-84.r.1.8, 168.144.1-84.r.1.1, 168.144.1-84.r.1.2, 168.144.1-84.r.1.3, 168.144.1-84.r.1.4, 168.144.1-84.r.1.5, 168.144.1-84.r.1.6, 168.144.1-84.r.1.7, 168.144.1-84.r.1.8 |
Cyclic 84-isogeny field degree: | $16$ |
Cyclic 84-torsion field degree: | $384$ |
Full 84-torsion field degree: | $129024$ |
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
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
6.36.0.a.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
84.24.0.q.1 | $84$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
84.36.0.c.1 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
84.36.1.ce.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
84.144.5.f.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.144.5.be.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.144.5.dk.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.144.5.do.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.144.5.fw.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.144.5.ge.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.144.5.gu.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.144.5.gz.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.dt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.ig.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.bbn.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.bcp.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.bsq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.buu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.bzc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.cal.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
252.216.7.by.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.216.7.ch.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.216.7.dh.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.216.7.dq.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.216.7.ee.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.216.7.en.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.216.10.bv.1 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
252.216.13.co.1 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |