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 | $4^{8}\cdot8^{2}$ | Cusp orbits | $2^{5}$ | ||
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: | 8N0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.0.272 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&24\\32&23\end{bmatrix}$, $\begin{bmatrix}11&4\\34&33\end{bmatrix}$, $\begin{bmatrix}15&36\\6&25\end{bmatrix}$, $\begin{bmatrix}27&36\\16&19\end{bmatrix}$, $\begin{bmatrix}31&12\\32&3\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 8.48.0.a.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 $ | $=$ | $ 2 x^{2} + y^{2} + 4 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 |
---|---|---|---|---|---|
40.48.0-4.b.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.48.0-4.b.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.48.0-8.d.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.48.0-8.d.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.48.0-8.d.2.5 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.48.0-8.d.2.11 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.