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 | $2^{4}\cdot8^{2}$ | 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: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.923 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&30\\38&17\end{bmatrix}$, $\begin{bmatrix}19&38\\11&15\end{bmatrix}$, $\begin{bmatrix}27&20\\12&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.0.t.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| 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 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6}{5^6}\cdot\frac{(5x+y)^{24}(15625x^{8}+625000x^{7}y-200000x^{6}y^{2}-2270000x^{5}y^{3}-1187000x^{4}y^{4}+436000x^{3}y^{5}+236800x^{2}y^{6}-6080xy^{7}-5744y^{8})^{3}}{(5x+y)^{24}(5x^{2}+2xy+2y^{2})^{8}(125x^{4}-500x^{3}y-600x^{2}y^{2}+40xy^{3}+44y^{4})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-20.d.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-20.d.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.240.8-40.bm.1.5 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.co.1.7 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.dk.1.15 | $40$ | $10$ | $10$ | $15$ |
| 120.144.4-120.fs.1.12 | $120$ | $3$ | $3$ | $4$ |
| 120.192.3-120.iq.1.13 | $120$ | $4$ | $4$ | $3$ |
| 280.384.11-280.dz.1.25 | $280$ | $8$ | $8$ | $11$ |