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: | 40.96.0.927 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&6\\16&3\end{bmatrix}$, $\begin{bmatrix}7&8\\36&31\end{bmatrix}$, $\begin{bmatrix}21&32\\24&25\end{bmatrix}$, $\begin{bmatrix}35&12\\12&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.o.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 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6\cdot5^2}\cdot\frac{(x-2y)^{48}(390625x^{16}+5000000x^{14}y^{2}+252000000x^{12}y^{4}+1523200000x^{10}y^{6}+4275200000x^{8}y^{8}+3899392000x^{6}y^{10}+1651507200x^{4}y^{12}+83886080x^{2}y^{14}+16777216y^{16})^{3}}{y^{4}x^{4}(x-2y)^{48}(5x^{2}-8y^{2})^{8}(5x^{2}+8y^{2})^{4}(25x^{4}+240x^{2}y^{2}+64y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.e.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.e.1.14 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.e.1.19 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.h.2.7 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.h.2.17 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.