Invariants
Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
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: | 12P1 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}31&44\\72&5\end{bmatrix}$, $\begin{bmatrix}53&27\\20&1\end{bmatrix}$, $\begin{bmatrix}75&74\\8&39\end{bmatrix}$, $\begin{bmatrix}77&9\\44&7\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 84.48.1.k.1 for the level structure with $-I$) |
Cyclic 84-isogeny field degree: | $8$ |
Cyclic 84-torsion field degree: | $192$ |
Full 84-torsion field degree: | $96768$ |
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 |
---|---|---|---|---|---|---|
12.48.0-12.g.1.5 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
84.24.0-28.g.1.2 | $84$ | $4$ | $4$ | $0$ | $?$ | full Jacobian |
84.48.0-12.g.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 |
---|---|---|---|---|---|---|
84.192.1-84.l.1.3 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
84.192.1-84.l.1.6 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
84.192.1-84.l.2.4 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
84.192.1-84.l.2.5 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
84.192.1-84.l.3.4 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
84.192.1-84.l.3.5 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
84.192.1-84.l.4.3 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
84.192.1-84.l.4.6 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
84.288.5-84.ca.1.7 | $84$ | $3$ | $3$ | $5$ | $?$ | not computed |
168.192.1-168.rp.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.1-168.rp.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.1-168.rp.2.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.1-168.rp.2.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.1-168.rp.3.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.1-168.rp.3.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.1-168.rp.4.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.1-168.rp.4.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.3-168.kq.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.kq.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.kq.2.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.kq.2.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.ks.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.ks.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.ks.2.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.ks.2.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.lo.1.17 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.lo.1.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.lp.1.37 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.lp.1.47 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.lw.1.25 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.lw.1.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.lx.1.17 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.lx.1.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.me.1.17 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.me.1.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.mf.1.25 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.mf.1.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.mi.1.25 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.mi.1.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.mj.1.17 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.mj.1.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.oi.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.oi.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.oi.2.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.oi.2.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.ok.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.ok.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.ok.2.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.192.3-168.ok.2.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
252.288.5-252.k.1.6 | $252$ | $3$ | $3$ | $5$ | $?$ | not computed |
252.288.9-252.s.1.7 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |
252.288.9-252.ba.1.3 | $252$ | $3$ | $3$ | $9$ | $?$ | not computed |