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 | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | X199f |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.96.0.165 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&3\\0&5\end{bmatrix}$, $\begin{bmatrix}3&3\\0&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $D_8$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.m.2 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $1$ |
| Cyclic 8-torsion field degree: | $2$ |
| 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 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2}\cdot\frac{(2x-y)^{48}(16x^{8}-128x^{7}y-416x^{6}y^{2}-448x^{5}y^{3}-360x^{4}y^{4}-224x^{3}y^{5}-104x^{2}y^{6}-16xy^{7}+y^{8})^{3}(16x^{8}+128x^{7}y-416x^{6}y^{2}+448x^{5}y^{3}-360x^{4}y^{4}+224x^{3}y^{5}-104x^{2}y^{6}+16xy^{7}+y^{8})^{3}}{y^{2}x^{2}(2x-y)^{48}(2x^{2}-y^{2})^{4}(2x^{2}+y^{2})^{2}(4x^{4}+12x^{2}y^{2}+y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.k.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.k.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.ba.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.ba.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.bb.2.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 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.