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 $4$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
| Rouse and Zureick-Brown (RZB) label: | X193i |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.96.0.57 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&2\\0&1\end{bmatrix}$, $\begin{bmatrix}1&6\\0&5\end{bmatrix}$, $\begin{bmatrix}7&6\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.l.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 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8}\cdot\frac{(4x-y)^{48}(16777216x^{16}-67108864x^{15}y+117440512x^{14}y^{2}-117440512x^{13}y^{3}+72351744x^{12}y^{4}-26214400x^{11}y^{5}+3276800x^{10}y^{6}+1900544x^{9}y^{7}-1183744x^{8}y^{8}+237568x^{7}y^{9}+51200x^{6}y^{10}-51200x^{5}y^{11}+17664x^{4}y^{12}-3584x^{3}y^{13}+448x^{2}y^{14}-32xy^{15}+y^{16})^{3}}{y^{8}x^{8}(2x-y)^{8}(4x-y)^{56}(8x^{2}-y^{2})^{2}(8x^{2}-8xy+y^{2})^{2}(8x^{2}-4xy+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.2.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.e.2.8 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.i.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.i.1.11 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.bb.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.48.0-8.bb.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.