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: | 40.48.0.610 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&20\\27&1\end{bmatrix}$, $\begin{bmatrix}7&32\\29&33\end{bmatrix}$, $\begin{bmatrix}9&4\\24&5\end{bmatrix}$, $\begin{bmatrix}13&32\\35&7\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.24.0.v.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 $ | $=$ | $ 2 x^{2} + 5 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 |
---|---|---|---|---|---|
8.24.0-4.d.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
20.24.0-4.d.1.1 | $20$ | $2$ | $2$ | $0$ | $0$ |
40.24.0-40.y.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.24.0-40.y.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.24.0-40.y.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.24.0-40.y.1.16 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
40.96.1-40.dg.1.3 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.dh.1.3 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.di.1.4 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.dj.1.3 | $40$ | $2$ | $2$ | $1$ |
40.240.8-40.bo.1.2 | $40$ | $5$ | $5$ | $8$ |
40.288.7-40.cq.1.3 | $40$ | $6$ | $6$ | $7$ |
40.480.15-40.dm.1.2 | $40$ | $10$ | $10$ | $15$ |
120.96.1-120.ki.1.5 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.kj.1.5 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.kk.1.5 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.kl.1.8 | $120$ | $2$ | $2$ | $1$ |
120.144.4-120.fw.1.9 | $120$ | $3$ | $3$ | $4$ |
120.192.3-120.jg.1.29 | $120$ | $4$ | $4$ | $3$ |
280.96.1-280.iu.1.5 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.iv.1.8 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.iw.1.8 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.ix.1.8 | $280$ | $2$ | $2$ | $1$ |
280.384.11-280.eh.1.10 | $280$ | $8$ | $8$ | $11$ |