Invariants
| Level: | $8$ | $\SL_2$-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 |
| Rouse and Zureick-Brown (RZB) label: | X33c |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.0.54 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&7\\0&5\end{bmatrix}$, $\begin{bmatrix}5&2\\0&3\end{bmatrix}$, $\begin{bmatrix}5&6\\4&7\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^2:\SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.m.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| Full 8-torsion field degree: | $64$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1248 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{x^{12}(4x^{4}+8x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{20}(8x^{2}+y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_1(4)$ | $4$ | $2$ | $2$ | $0$ | $0$ |
| 8.12.0-4.c.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.