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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.0.1059 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&32\\33&23\end{bmatrix}$, $\begin{bmatrix}21&16\\29&7\end{bmatrix}$, $\begin{bmatrix}25&32\\11&3\end{bmatrix}$, $\begin{bmatrix}39&8\\9&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.n.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $48$ |
| 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 12 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{x^{48}(x^{8}-16x^{6}y^{2}+8x^{4}y^{4}+64x^{2}y^{6}+16y^{8})^{3}(x^{8}+16x^{6}y^{2}+8x^{4}y^{4}-64x^{2}y^{6}+16y^{8})^{3}}{y^{4}x^{52}(x^{2}-2y^{2})^{2}(x^{2}+2y^{2})^{2}(x^{2}-2xy+2y^{2})^{8}(x^{2}+2xy+2y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.q.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.q.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.ba.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.ba.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.bb.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.bb.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.