Invariants
Level: | $168$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{4}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8F1 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}1&136\\165&95\end{bmatrix}$, $\begin{bmatrix}21&152\\121&7\end{bmatrix}$, $\begin{bmatrix}49&68\\141&23\end{bmatrix}$, $\begin{bmatrix}167&28\\147&25\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 24.48.1.dy.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $3072$ |
Full 168-torsion field degree: | $1548288$ |
Jacobian
Conductor: | $?$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 y z - 3 z^{2} + 2 w^{2} $ |
$=$ | $2 x^{2} + 2 x y - 4 x z + y^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} - 6 x^{3} y + 2 x^{2} y^{2} - 12 x^{2} z^{2} - 4 x y z^{2} + 4 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^2}\cdot\frac{(3y^{2}-4w^{2})^{3}(3y^{2}+4w^{2})^{3}}{w^{8}y^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.dy.1 :
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 3x$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
$0$ | $=$ | $ 9X^{4}-6X^{3}Y+2X^{2}Y^{2}-12X^{2}Z^{2}-4XYZ^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.48.0-4.c.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
168.48.0-4.c.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
168.48.0-24.u.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
168.48.0-24.u.1.6 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
168.48.1-24.n.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1-24.n.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | 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 |
---|---|---|---|---|---|---|
168.288.9-24.yy.1.4 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
168.384.9-24.ja.1.2 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |