Invariants
Level: | $168$ | $\SL_2$-level: | $4$ | ||||
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 | $2^{2}\cdot4^{2}$ | 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: | 4E0 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}45&70\\4&155\end{bmatrix}$, $\begin{bmatrix}91&144\\144&95\end{bmatrix}$, $\begin{bmatrix}95&24\\78&67\end{bmatrix}$, $\begin{bmatrix}97&54\\74&139\end{bmatrix}$, $\begin{bmatrix}127&134\\114&83\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 84.12.0.a.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $128$ |
Cyclic 168-torsion field degree: | $6144$ |
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 |
---|---|---|---|---|---|
8.12.0-2.a.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
168.12.0-2.a.1.1 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.48.0-84.a.1.3 | $168$ | $2$ | $2$ | $0$ |
168.48.0-84.a.1.4 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.b.1.9 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.b.1.12 | $168$ | $2$ | $2$ | $0$ |
168.48.0-84.c.1.10 | $168$ | $2$ | $2$ | $0$ |
168.48.0-84.c.1.15 | $168$ | $2$ | $2$ | $0$ |
168.48.0-84.d.1.5 | $168$ | $2$ | $2$ | $0$ |
168.48.0-84.d.1.8 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.f.1.9 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.f.1.12 | $168$ | $2$ | $2$ | $0$ |
168.48.0-84.g.1.3 | $168$ | $2$ | $2$ | $0$ |
168.48.0-84.g.1.7 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.j.1.10 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.j.1.11 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.s.1.10 | $168$ | $2$ | $2$ | $0$ |
168.48.0-168.s.1.11 | $168$ | $2$ | $2$ | $0$ |
168.72.2-84.c.1.4 | $168$ | $3$ | $3$ | $2$ |
168.96.1-84.c.1.8 | $168$ | $4$ | $4$ | $1$ |
168.192.5-84.c.1.7 | $168$ | $8$ | $8$ | $5$ |
168.504.16-84.c.1.14 | $168$ | $21$ | $21$ | $16$ |