Invariants
| Level: | $40$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.12.0.54 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&30\\27&27\end{bmatrix}$, $\begin{bmatrix}7&0\\35&37\end{bmatrix}$, $\begin{bmatrix}21&22\\12&17\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $61440$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ x^{2} - 10 y^{2} - 320 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
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.6.0.f.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 20.6.0.c.1 | $20$ | $2$ | $2$ | $0$ | $0$ |
| 40.6.0.b.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.60.4.cc.1 | $40$ | $5$ | $5$ | $4$ |
| 40.72.3.ec.1 | $40$ | $6$ | $6$ | $3$ |
| 40.120.7.gc.1 | $40$ | $10$ | $10$ | $7$ |
| 120.36.2.qu.1 | $120$ | $3$ | $3$ | $2$ |
| 120.48.1.bzs.1 | $120$ | $4$ | $4$ | $1$ |
| 280.96.5.fi.1 | $280$ | $8$ | $8$ | $5$ |
| 280.252.16.kg.1 | $280$ | $21$ | $21$ | $16$ |
| 280.336.21.kg.1 | $280$ | $28$ | $28$ | $21$ |