Invariants
Level: | $168$ | $\SL_2$-level: | $8$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{4}\cdot2$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G0 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}17&8\\80&129\end{bmatrix}$, $\begin{bmatrix}31&8\\8&167\end{bmatrix}$, $\begin{bmatrix}91&64\\116&27\end{bmatrix}$, $\begin{bmatrix}119&88\\40&1\end{bmatrix}$, $\begin{bmatrix}131&16\\48&167\end{bmatrix}$, $\begin{bmatrix}143&72\\66&47\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 8.24.0.i.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $32$ |
Cyclic 168-torsion field degree: | $1536$ |
Full 168-torsion field degree: | $3096576$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 122 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{8}+240x^{6}y^{2}+2144x^{4}y^{4}+3840x^{2}y^{6}+256y^{8})^{3}}{y^{2}x^{26}(x-2y)^{8}(x+2y)^{8}(x^{2}+4y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
168.24.0-4.b.1.6 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-4.b.1.9 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.96.0-8.j.1.4 | $168$ | $2$ | $2$ | $0$ |
168.96.0-8.j.2.5 | $168$ | $2$ | $2$ | $0$ |
168.96.0-8.k.1.5 | $168$ | $2$ | $2$ | $0$ |
168.96.0-8.k.1.6 | $168$ | $2$ | $2$ | $0$ |
168.96.0-8.k.2.6 | $168$ | $2$ | $2$ | $0$ |
168.96.0-8.k.2.8 | $168$ | $2$ | $2$ | $0$ |
168.96.0-8.l.1.3 | $168$ | $2$ | $2$ | $0$ |
168.96.0-8.l.2.6 | $168$ | $2$ | $2$ | $0$ |
168.96.0-56.z.1.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-56.z.2.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-24.ba.1.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-24.ba.2.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-56.ba.1.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-56.ba.1.13 | $168$ | $2$ | $2$ | $0$ |
168.96.0-56.ba.2.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-56.ba.2.13 | $168$ | $2$ | $2$ | $0$ |
168.96.0-24.bb.1.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-24.bb.1.11 | $168$ | $2$ | $2$ | $0$ |
168.96.0-24.bb.2.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-24.bb.2.13 | $168$ | $2$ | $2$ | $0$ |
168.96.0-56.bb.1.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-56.bb.2.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-24.bc.1.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-24.bc.2.9 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.cw.1.17 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.cw.2.23 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.cx.1.23 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.cx.1.25 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.cx.2.19 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.cx.2.29 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.cy.1.17 | $168$ | $2$ | $2$ | $0$ |
168.96.0-168.cy.2.21 | $168$ | $2$ | $2$ | $0$ |
168.96.1-8.h.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-8.h.1.10 | $168$ | $2$ | $2$ | $1$ |
168.96.1-8.p.1.2 | $168$ | $2$ | $2$ | $1$ |
168.96.1-8.p.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-24.bu.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-24.bu.1.9 | $168$ | $2$ | $2$ | $1$ |
168.96.1-56.bu.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-56.bu.1.10 | $168$ | $2$ | $2$ | $1$ |
168.96.1-24.bv.1.2 | $168$ | $2$ | $2$ | $1$ |
168.96.1-24.bv.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-56.bv.1.2 | $168$ | $2$ | $2$ | $1$ |
168.96.1-56.bv.1.10 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.fq.1.12 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.fq.1.18 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.fr.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.fr.1.20 | $168$ | $2$ | $2$ | $1$ |
168.144.4-24.ch.1.16 | $168$ | $3$ | $3$ | $4$ |
168.192.3-24.cl.1.13 | $168$ | $4$ | $4$ | $3$ |
168.384.11-56.bn.1.43 | $168$ | $8$ | $8$ | $11$ |