Invariants
Level: | $168$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 96$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8K1 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}23&144\\134&139\end{bmatrix}$, $\begin{bmatrix}133&12\\130&151\end{bmatrix}$, $\begin{bmatrix}139&16\\130&3\end{bmatrix}$, $\begin{bmatrix}145&156\\148&43\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.96.1.ce.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $32$ |
Cyclic 168-torsion field degree: | $1536$ |
Full 168-torsion field degree: | $774144$ |
Jacobian
Conductor: | $?$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 3136.2.a.m |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + y^{2} + z^{2} $ |
$=$ | $14 y^{2} - 14 z^{2} + w^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{7^4}\cdot\frac{(38416z^{8}-5488z^{6}w^{2}+980z^{4}w^{4}-56z^{2}w^{6}+w^{8})^{3}}{w^{4}z^{8}(14z^{2}-w^{2})^{4}(28z^{2}-w^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
24.96.0-8.j.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
168.96.0-8.j.1.4 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
168.96.0-56.k.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
168.96.0-56.k.1.10 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
168.96.0-56.l.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
168.96.0-56.l.1.10 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
168.96.0-56.ba.1.1 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
168.96.0-56.ba.1.9 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
168.96.1-56.bg.2.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.96.1-56.bg.2.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.96.1-56.bh.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.96.1-56.bh.1.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.96.1-56.bv.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.96.1-56.bv.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |