Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| 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: | 8O0 |
| Rouse and Zureick-Brown (RZB) label: | X202h |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.96.0.157 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&5\\0&5\end{bmatrix}$, $\begin{bmatrix}1&7\\0&3\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $\SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.p.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $1$ |
| Cyclic 8-torsion field degree: | $1$ |
| Full 8-torsion field degree: | $16$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3}\cdot\frac{(4x-y)^{48}(16777216x^{16}+486539264x^{14}y^{2}+191889408x^{12}y^{4}-39059456x^{10}y^{6}+2908160x^{8}y^{8}-610304x^{6}y^{10}+46848x^{4}y^{12}+1856x^{2}y^{14}+y^{16})^{3}}{y^{2}x^{2}(4x-y)^{48}(8x^{2}-y^{2})^{2}(8x^{2}+y^{2})^{4}(8x^{2}-8xy+y^{2})^{8}(8x^{2}+8xy+y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.r.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.r.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| $X_1(8)$ | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.bb.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.bb.2.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.bb.2.8 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.