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 | $2^{2}\cdot4^{3}\cdot8$ | 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: | 8J0 |
Rouse and Zureick-Brown (RZB) label: | X99m |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.16 |
Level structure
$\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&0\\4&5\end{bmatrix}$, $\begin{bmatrix}1&4\\0&5\end{bmatrix}$, $\begin{bmatrix}5&4\\0&7\end{bmatrix}$, $\begin{bmatrix}7&2\\4&5\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: | $C_2^2\times D_4$ |
Contains $-I$: | no $\quad$ (see 8.24.0.d.1 for the level structure with $-I$) |
Cyclic 8-isogeny field degree: | $2$ |
Cyclic 8-torsion field degree: | $8$ |
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 136 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{x^{24}(256x^{8}+256x^{6}y^{2}+80x^{4}y^{4}+8x^{2}y^{6}+y^{8})^{3}}{y^{8}x^{28}(2x^{2}+y^{2})^{2}(4x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
4.24.0-4.b.1.2 | $4$ | $2$ | $2$ | $0$ | $0$ |
8.24.0-4.b.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.