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: | $10$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8N0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.0.803 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&18\\4&5\end{bmatrix}$, $\begin{bmatrix}7&34\\32&15\end{bmatrix}$, $\begin{bmatrix}7&36\\4&29\end{bmatrix}$, $\begin{bmatrix}29&10\\8&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.p.2 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
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{10}\cdot5^2}\cdot\frac{x^{48}(390625x^{16}-20000000x^{14}y^{2}+4032000000x^{12}y^{4}-97484800000x^{10}y^{6}+1094451200000x^{8}y^{8}-3992977408000x^{6}y^{10}+6764573491200x^{4}y^{12}-1374389534720x^{2}y^{14}+1099511627776y^{16})^{3}}{y^{4}x^{52}(5x^{2}-32y^{2})^{4}(5x^{2}+32y^{2})^{8}(25x^{4}-960x^{2}y^{2}+1024y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.1.10 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.e.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.e.1.17 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.i.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.i.1.28 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.