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: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
| Rouse and Zureick-Brown (RZB) label: | X190e |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.96.0.113 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}5&4\\4&7\end{bmatrix}$, $\begin{bmatrix}5&6\\4&1\end{bmatrix}$, $\begin{bmatrix}7&6\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.h.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| 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 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(x-y)^{48}(x^{8}-4x^{7}y+16x^{6}y^{2}-56x^{5}y^{3}+120x^{4}y^{4}-112x^{3}y^{5}+64x^{2}y^{6}-32xy^{7}+16y^{8})^{3}(13x^{8}-140x^{7}y+688x^{6}y^{2}-1960x^{5}y^{3}+3480x^{4}y^{4}-3920x^{3}y^{5}+2752x^{2}y^{6}-1120xy^{7}+208y^{8})^{3}}{(x-y)^{48}(x^{2}-2y^{2})^{4}(x^{2}-4xy+2y^{2})^{8}(x^{2}-4xy+6y^{2})^{2}(x^{2}-2xy+2y^{2})^{8}(3x^{2}-4xy+2y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.d.1.13 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.d.1.16 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.e.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.e.1.7 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.h.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.h.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.