Invariants
| Level: | $40$ | $\SL_2$-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 | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.0.924 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&12\\16&15\end{bmatrix}$, $\begin{bmatrix}11&36\\22&15\end{bmatrix}$, $\begin{bmatrix}15&32\\32&13\end{bmatrix}$, $\begin{bmatrix}21&16\\14&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.w.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $7680$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 10 x^{2} + 4 y^{2} + 2 y z - z^{2} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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.e.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.h.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.h.1.15 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.m.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.m.1.19 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.