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 | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.0.515 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&0\\28&31\end{bmatrix}$, $\begin{bmatrix}3&12\\24&23\end{bmatrix}$, $\begin{bmatrix}11&32\\14&19\end{bmatrix}$, $\begin{bmatrix}13&12\\16&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.48.0.h.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 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^8}\cdot\frac{(x+2y)^{48}(13x^{8}-8x^{7}y+1024x^{6}y^{2}+9280x^{5}y^{3}+27520x^{4}y^{4}-52736x^{3}y^{5}+237568x^{2}y^{6}+815104xy^{7}+1724416y^{8})^{3}(421x^{8}-1592x^{7}y+3712x^{6}y^{2}+6592x^{5}y^{3}+27520x^{4}y^{4}-74240x^{3}y^{5}+65536x^{2}y^{6}+4096xy^{7}+53248y^{8})^{3}}{(x-4y)^{8}(x+2y)^{56}(x^{2}+8y^{2})^{8}(x^{2}+16xy-8y^{2})^{4}(17x^{4}-32x^{3}y+48x^{2}y^{2}+256xy^{3}+1088y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.48.0-8.d.2.12 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.d.2.14 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.2.10 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.2.14 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.h.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.h.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.