Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 8I0 |
| Rouse and Zureick-Brown (RZB) label: | X86n |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.184 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&7\\0&7\end{bmatrix}$, $\begin{bmatrix}3&0\\0&5\end{bmatrix}$, $\begin{bmatrix}5&5\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $D_8:C_2$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.ba.2 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $1$ |
| Cyclic 8-torsion field degree: | $4$ |
| Full 8-torsion field degree: | $32$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 131 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2}{3^4}\cdot\frac{(6x-y)^{24}(78941952x^{8}-342641664x^{7}y+528705792x^{6}y^{2}-425440512x^{5}y^{3}+202292640x^{4}y^{4}-58888512x^{3}y^{5}+10248336x^{2}y^{6}-966192xy^{7}+37199y^{8})^{3}}{(2x-y)^{4}(6x-y)^{26}(36x^{2}-24xy+5y^{2})(108x^{2}-60xy+11y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0-8.n.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 8.24.0-8.n.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.