Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | ||||
Index: | $24$ | $\PSL_2$-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: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.0.292 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&38\\24&3\end{bmatrix}$, $\begin{bmatrix}21&4\\18&31\end{bmatrix}$, $\begin{bmatrix}31&39\\18&11\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 8.12.0.c.1 for the level structure with $-I$) |
Cyclic 40-isogeny field degree: | $24$ |
Cyclic 40-torsion field degree: | $384$ |
Full 40-torsion field degree: | $30720$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 2 x^{2} + y^{2} + 64 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 |
---|---|---|---|---|---|
40.12.0-4.b.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.12.0-4.b.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
120.72.2-24.i.1.3 | $120$ | $3$ | $3$ | $2$ |
120.96.1-24.cf.1.12 | $120$ | $4$ | $4$ | $1$ |
40.120.4-40.e.1.3 | $40$ | $5$ | $5$ | $4$ |
40.144.3-40.g.1.7 | $40$ | $6$ | $6$ | $3$ |
40.240.7-40.i.1.7 | $40$ | $10$ | $10$ | $7$ |
280.192.5-56.e.1.12 | $280$ | $8$ | $8$ | $5$ |
280.504.16-56.i.1.6 | $280$ | $21$ | $21$ | $16$ |