Invariants
Level: | $168$ | $\SL_2$-level: | $8$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{2}\cdot2$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8C0 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}8&43\\111&92\end{bmatrix}$, $\begin{bmatrix}71&140\\106&153\end{bmatrix}$, $\begin{bmatrix}115&10\\68&13\end{bmatrix}$, $\begin{bmatrix}129&88\\134&39\end{bmatrix}$, $\begin{bmatrix}163&146\\64&9\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.12.0.y.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $1536$ |
Full 168-torsion field degree: | $6193152$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.12.0-4.c.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
168.12.0-4.c.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.48.0-168.x.1.17 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.z.1.16 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.bf.1.6 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.bh.1.12 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.bv.1.9 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.bw.1.2 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.by.1.11 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.cb.1.6 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.ck.1.9 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.cn.1.10 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.cp.1.9 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.cq.1.14 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.da.1.12 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.dd.1.16 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.dr.1.9 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.ds.1.4 | $168$ | $2$ | $2$ | $0$ |
168.72.2-168.cw.1.23 | $168$ | $3$ | $3$ | $2$ |
168.96.1-168.zu.1.12 | $168$ | $4$ | $4$ | $1$ |
168.192.5-168.fy.1.22 | $168$ | $8$ | $8$ | $5$ |
168.504.16-168.cw.1.51 | $168$ | $21$ | $21$ | $16$ |