Invariants
Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12V1 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}31&66\\64&59\end{bmatrix}$, $\begin{bmatrix}43&24\\30&1\end{bmatrix}$, $\begin{bmatrix}61&6\\52&83\end{bmatrix}$, $\begin{bmatrix}61&78\\76&13\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 84.96.1.f.4 for the level structure with $-I$) |
Cyclic 84-isogeny field degree: | $16$ |
Cyclic 84-torsion field degree: | $192$ |
Full 84-torsion field degree: | $48384$ |
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 |
---|---|---|---|---|---|---|
12.96.0-12.a.1.15 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
84.96.0-12.a.1.5 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
84.96.0-84.a.1.21 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
84.96.0-84.a.1.24 | $84$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
84.96.1-84.b.1.6 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
84.96.1-84.b.1.8 | $84$ | $2$ | $2$ | $1$ | $?$ | 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 |
---|---|---|---|---|---|---|
84.384.5-84.g.2.3 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.384.5-84.h.1.6 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.384.5-84.m.2.5 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.384.5-84.o.4.5 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.384.5-84.q.1.5 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.384.5-84.r.3.5 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.384.5-84.w.3.6 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.384.5-84.y.4.8 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-168.iu.4.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-168.ja.4.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-168.ke.3.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-168.ks.3.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-168.og.3.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-168.om.3.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-168.pq.4.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.384.5-168.qe.4.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |