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: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G0 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}93&44\\151&77\end{bmatrix}$, $\begin{bmatrix}107&136\\29&59\end{bmatrix}$, $\begin{bmatrix}129&128\\152&111\end{bmatrix}$, $\begin{bmatrix}141&44\\14&71\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.24.0.bn.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $3072$ |
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 26 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{7}\cdot\frac{(x+4y)^{24}(x^{4}+224x^{3}y+2464x^{2}y^{2}-37632y^{4})^{3}(3x^{4}-2464x^{2}y^{2}+25088xy^{3}-12544y^{4})^{3}}{(x+4y)^{24}(x^{2}+112y^{2})^{8}(x^{2}-8xy-112y^{2})^{2}(x^{2}+56xy-112y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.24.0-8.p.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ |
84.24.0-28.g.1.2 | $84$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-28.g.1.6 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-8.p.1.8 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-56.y.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ |
168.24.0-56.y.1.16 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.144.4-168.kb.1.30 | $168$ | $3$ | $3$ | $4$ |
168.192.3-168.mj.1.29 | $168$ | $4$ | $4$ | $3$ |
168.384.11-56.ep.1.2 | $168$ | $8$ | $8$ | $11$ |