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 $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
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: | 8O0 |
Rouse and Zureick-Brown (RZB) label: | X185g |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.96.0.41 |
Level structure
$\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&4\\0&3\end{bmatrix}$, $\begin{bmatrix}5&0\\0&1\end{bmatrix}$, $\begin{bmatrix}5&6\\0&5\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: | $C_2\times D_4$ |
Contains $-I$: | no $\quad$ (see 8.48.0.k.1 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 12 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{x^{48}(x^{16}+240x^{12}y^{4}+2144x^{8}y^{8}+3840x^{4}y^{12}+256y^{16})^{3}}{y^{4}x^{52}(x^{2}-2y^{2})^{8}(x^{2}+2y^{2})^{8}(x^{2}-2xy+2y^{2})^{2}(x^{2}+2xy+2y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.48.0-8.d.2.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.48.0-8.d.2.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.48.0-8.e.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.48.0-8.e.1.9 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.48.0-8.i.1.8 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.48.0-8.i.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.