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 | $4^{8}\cdot8^{2}$ | 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: | 8N0 |
| Rouse and Zureick-Brown (RZB) label: | X204d |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.96.0.137 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}3&2\\4&7\end{bmatrix}$, $\begin{bmatrix}5&2\\0&3\end{bmatrix}$, $\begin{bmatrix}7&0\\4&7\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.e.2 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 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{(2x+y)^{48}(18688x^{16}+90112x^{15}y+216064x^{14}y^{2}+372736x^{13}y^{3}+575232x^{12}y^{4}+845824x^{11}y^{5}+1094912x^{10}y^{6}+1143808x^{9}y^{7}+924512x^{8}y^{8}+571904x^{7}y^{9}+273728x^{6}y^{10}+105728x^{5}y^{11}+35952x^{4}y^{12}+11648x^{3}y^{13}+3376x^{2}y^{14}+704xy^{15}+73y^{16})^{3}}{(2x+y)^{48}(2x^{2}-y^{2})^{4}(2x^{2}+y^{2})^{4}(2x^{2}+2xy+y^{2})^{4}(2x^{2}+4xy+y^{2})^{8}(6x^{2}+8xy+3y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.c.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.c.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.d.2.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.d.2.9 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.e.2.8 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.e.2.12 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.