Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
| 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: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.846 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}13&4\\11&21\end{bmatrix}$, $\begin{bmatrix}15&8\\28&21\end{bmatrix}$, $\begin{bmatrix}33&24\\3&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.0.n.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $15360$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ x^{2} + 10 y^{2} + 80 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 |
|---|---|---|---|---|---|
| 8.24.0-8.d.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-8.d.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-40.y.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-40.y.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-40.ba.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-40.ba.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.240.8-40.x.1.1 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.bt.1.5 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.cj.1.1 | $40$ | $10$ | $10$ | $15$ |
| 120.144.4-120.eu.1.1 | $120$ | $3$ | $3$ | $4$ |
| 120.192.3-120.hq.1.1 | $120$ | $4$ | $4$ | $3$ |
| 280.384.11-280.de.1.5 | $280$ | $8$ | $8$ | $11$ |