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: | $6$ (of which $4$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.53 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&4\\24&29\end{bmatrix}$, $\begin{bmatrix}27&12\\6&29\end{bmatrix}$, $\begin{bmatrix}29&32\\0&33\end{bmatrix}$, $\begin{bmatrix}37&8\\12&25\end{bmatrix}$, $\begin{bmatrix}39&20\\24&11\end{bmatrix}$, $\begin{bmatrix}39&24\\30&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.24.0.b.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| 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 61 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{4}-4x^{3}y+8x^{2}y^{2}+16xy^{3}+16y^{4})^{3}(x^{4}+4x^{3}y+8x^{2}y^{2}-16xy^{3}+16y^{4})^{3}}{y^{4}x^{28}(x-2y)^{4}(x+2y)^{4}(x^{2}+4y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-4.a.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-4.a.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.96.0-8.a.1.8 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.a.1.15 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.b.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.b.2.6 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.b.1.8 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.b.2.8 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.c.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.c.1.15 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-8.g.1.7 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-8.g.2.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-8.h.1.3 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-8.h.2.4 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.n.1.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.n.2.21 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.o.1.6 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.o.2.17 | $40$ | $2$ | $2$ | $1$ |
| 40.96.2-8.a.1.1 | $40$ | $2$ | $2$ | $2$ |
| 40.96.2-8.a.1.11 | $40$ | $2$ | $2$ | $2$ |
| 40.96.2-40.a.1.14 | $40$ | $2$ | $2$ | $2$ |
| 40.96.2-40.a.1.20 | $40$ | $2$ | $2$ | $2$ |
| 40.240.8-20.b.1.19 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-20.b.1.14 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-20.b.1.6 | $40$ | $10$ | $10$ | $15$ |
| 120.96.0-24.a.1.17 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.a.1.29 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.b.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.b.2.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.b.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.b.2.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.c.1.20 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.c.1.31 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-24.n.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.n.2.17 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.n.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.n.2.38 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.o.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.o.2.21 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.o.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.o.2.38 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-24.a.1.13 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-24.a.1.19 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.a.1.19 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.a.1.40 | $120$ | $2$ | $2$ | $2$ |
| 120.144.4-12.b.1.14 | $120$ | $3$ | $3$ | $4$ |
| 120.192.3-12.b.1.32 | $120$ | $4$ | $4$ | $3$ |
| 280.96.0-56.a.1.16 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.a.1.39 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.b.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.b.2.5 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.b.1.10 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.b.2.12 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.c.1.19 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.c.1.31 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-56.n.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.n.2.21 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.n.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.n.2.44 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.o.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.o.2.21 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.o.1.9 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.o.2.36 | $280$ | $2$ | $2$ | $1$ |
| 280.96.2-56.a.1.9 | $280$ | $2$ | $2$ | $2$ |
| 280.96.2-56.a.1.11 | $280$ | $2$ | $2$ | $2$ |
| 280.96.2-280.a.1.23 | $280$ | $2$ | $2$ | $2$ |
| 280.96.2-280.a.1.38 | $280$ | $2$ | $2$ | $2$ |
| 280.384.11-28.b.1.40 | $280$ | $8$ | $8$ | $11$ |