Invariants
| Level: | $40$ | $\SL_2$-level: | $4$ | ||||
| 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 | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
| 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: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.91 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&10\\36&29\end{bmatrix}$, $\begin{bmatrix}9&22\\18&17\end{bmatrix}$, $\begin{bmatrix}35&16\\14&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.0.d.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $15360$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 40 x^{2} - y^{2} - 3 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 |
|---|---|---|---|---|---|
| 8.24.0-8.a.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 20.24.0-20.b.1.4 | $20$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-8.a.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-20.b.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-40.b.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-40.b.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ |
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.g.1.4 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.g.1.6 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.g.1.8 | $40$ | $10$ | $10$ | $15$ |
| 120.144.4-120.d.1.14 | $120$ | $3$ | $3$ | $4$ |
| 120.192.3-120.dr.1.19 | $120$ | $4$ | $4$ | $3$ |
| 280.384.11-280.d.1.22 | $280$ | $8$ | $8$ | $11$ |