Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.877 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}19&20\\3&27\end{bmatrix}$, $\begin{bmatrix}19&22\\12&37\end{bmatrix}$, $\begin{bmatrix}29&16\\13&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.24.0.e.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $15360$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 22 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^{14}}{5^3}\cdot\frac{(x+2y)^{24}(31x^{8}-11x^{7}y+7x^{6}y^{2}+77x^{5}y^{3}+420x^{4}y^{4}-77x^{3}y^{5}+7x^{2}y^{6}+11xy^{7}+31y^{8})^{3}}{(x+2y)^{24}(x^{2}+y^{2})^{4}(x^{4}+22x^{3}y-6x^{2}y^{2}-22xy^{3}+y^{4})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-20.e.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-20.e.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.240.8-20.h.1.7 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-20.s.1.13 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-20.t.1.3 | $40$ | $10$ | $10$ | $15$ |
| 120.144.4-60.bo.1.28 | $120$ | $3$ | $3$ | $4$ |
| 120.192.3-60.bk.1.11 | $120$ | $4$ | $4$ | $3$ |
| 280.384.11-140.bk.1.9 | $280$ | $8$ | $8$ | $11$ |