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: | 40.96.0.200 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&16\\14&23\end{bmatrix}$, $\begin{bmatrix}7&24\\32&11\end{bmatrix}$, $\begin{bmatrix}31&8\\12&35\end{bmatrix}$, $\begin{bmatrix}37&12\\28&9\end{bmatrix}$, $\begin{bmatrix}37&32\\20&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.b.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| 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 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{5^4}\cdot\frac{(x+y)^{48}(64x^{8}+1280x^{7}y+11040x^{6}y^{2}+53600x^{5}y^{3}+160200x^{4}y^{4}+302000x^{3}y^{5}+351500x^{2}y^{6}+232500xy^{7}+68125y^{8})^{3}(64x^{8}+1280x^{7}y+11360x^{6}y^{2}+58400x^{5}y^{3}+190200x^{4}y^{4}+402000x^{3}y^{5}+538500x^{2}y^{6}+417500xy^{7}+143125y^{8})^{3}}{y^{8}(x+y)^{48}(2x+5y)^{8}(x^{2}+5xy+5y^{2})^{4}(2x^{2}+10xy+15y^{2})^{4}(8x^{4}+80x^{3}y+300x^{2}y^{2}+500xy^{3}+325y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-4.b.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-4.b.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.h.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.h.1.19 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.i.2.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.i.2.25 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.