Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | ||||
Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
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: | 8O0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.0.587 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}17&0\\36&13\end{bmatrix}$, $\begin{bmatrix}19&12\\26&1\end{bmatrix}$, $\begin{bmatrix}33&28\\18&9\end{bmatrix}$, $\begin{bmatrix}39&32\\38&15\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.48.0.u.1 for the level structure with $-I$) |
Cyclic 40-isogeny field degree: | $12$ |
Cyclic 40-torsion field degree: | $192$ |
Full 40-torsion field degree: | $7680$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 5 x^{2} + 40 y^{2} + 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 |
---|---|---|---|---|---|
8.48.0-8.d.2.13 | $8$ | $2$ | $2$ | $0$ | $0$ |
40.48.0-8.d.2.15 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.48.0-40.h.1.22 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.48.0-40.h.1.23 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.48.0-40.m.1.17 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.48.0-40.m.1.18 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.