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: | 8G1 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}29&8\\12&37\end{bmatrix}$, $\begin{bmatrix}31&92\\48&55\end{bmatrix}$, $\begin{bmatrix}131&136\\80&23\end{bmatrix}$, $\begin{bmatrix}147&134\\32&83\end{bmatrix}$, $\begin{bmatrix}153&110\\16&99\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.1.bh.2 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 x y + 3 x z - y^{2} + 2 y z - z^{2} $ |
| $=$ | $2 x^{2} + 6 y^{2} + 4 y z + 6 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 7 x^{4} + 17 x^{3} z + 2 x^{2} y^{2} + 24 x^{2} z^{2} + 4 x y^{2} z + 17 x z^{3} + 2 y^{2} z^{2} + 7 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{7^4}\cdot\frac{26842747207680xz^{11}+35656189599744xz^{9}w^{2}+13402161206784xz^{7}w^{4}+5700739817472xz^{5}w^{6}+7584075060744xz^{3}w^{8}-65850275758080y^{2}z^{10}-77114746662912y^{2}z^{8}w^{2}-52452992675328y^{2}z^{6}w^{4}+2293840104640y^{2}z^{4}w^{6}+10492896721776y^{2}z^{2}w^{8}+2659434619443y^{2}w^{10}-39007528550400yz^{11}-13227260350464yz^{9}w^{2}-31689108777984yz^{7}w^{4}+1662512087488yz^{5}w^{6}+14921322195696yz^{3}w^{8}+1220785667241yzw^{10}-11239081672704z^{12}+17152930160640z^{10}w^{2}-12870075563520z^{8}w^{4}+1872729773824z^{6}w^{6}+16113624874368z^{4}w^{8}+5503462767222z^{2}w^{10}+433881982464w^{12}}{w^{4}(5013504xz^{7}-688128xz^{5}w^{2}+130536xz^{3}w^{4}-11042816y^{2}z^{6}+5562368y^{2}z^{4}w^{2}+244608y^{2}z^{2}w^{4}-9261y^{2}w^{6}-6029312yz^{7}+2367488yz^{5}w^{2}+906192yz^{3}w^{4}+30429yzw^{6}-6029312z^{8}+733184z^{6}w^{2}+454608z^{4}w^{4}+39690z^{2}w^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.bh.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{3}{4}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 7X^{4}+2X^{2}Y^{2}+17X^{3}Z+4XY^{2}Z+24X^{2}Z^{2}+2Y^{2}Z^{2}+17XZ^{3}+7Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.48.0-8.d.2.5 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 168.48.0-8.d.2.13 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.48.0-24.i.2.3 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.48.0-24.i.2.27 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.48.1-24.d.1.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1-24.d.1.15 | $168$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 168.192.1-24.a.2.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-24.r.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-24.bj.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-24.bn.2.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-24.bs.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-24.bw.2.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-24.ce.2.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-24.cg.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ha.2.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.he.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.hq.1.9 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.hu.2.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.jm.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.jq.2.13 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.kc.2.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.kg.1.9 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.288.9-24.ha.1.17 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.384.9-24.ef.2.13 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |