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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8O0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.96.0.163 |
Level structure
$\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}5&3\\0&1\end{bmatrix}$, $\begin{bmatrix}7&7\\0&5\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: | $\SD_{16}$ |
Contains $-I$: | no $\quad$ (see 8.48.0.o.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
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ x^{2} + 2 y^{2} + 8 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.48.0-8.r.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.48.0-8.r.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.48.0-8.ba.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.48.0-8.ba.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.48.0-8.ba.2.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.48.0-8.ba.2.8 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.