Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.0.34 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}15&2\\36&9\end{bmatrix}$, $\begin{bmatrix}31&22\\10&21\end{bmatrix}$, $\begin{bmatrix}35&2\\4&9\end{bmatrix}$, $\begin{bmatrix}37&26\\18&15\end{bmatrix}$, $\begin{bmatrix}39&18\\26&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.12.0.a.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $30720$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 746 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{2}-4xy+16y^{2})^{3}(x^{2}+4xy+16y^{2})^{3}}{y^{4}x^{16}(x^{2}+16y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.12.0-4.b.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.12.0-4.b.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.48.0-4.a.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-4.b.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.a.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.b.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.f.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.g.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.1-8.a.1.4 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1-8.b.1.2 | $40$ | $2$ | $2$ | $1$ |
| 120.48.0-12.a.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-12.b.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.72.2-12.a.1.21 | $120$ | $3$ | $3$ | $2$ |
| 120.96.1-12.a.1.19 | $120$ | $4$ | $4$ | $1$ |
| 40.48.0-20.a.1.6 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-20.b.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.120.4-20.a.1.12 | $40$ | $5$ | $5$ | $4$ |
| 40.144.3-20.a.1.11 | $40$ | $6$ | $6$ | $3$ |
| 40.240.7-20.a.1.22 | $40$ | $10$ | $10$ | $7$ |
| 120.48.0-24.a.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.c.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.j.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.k.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1-24.a.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-24.b.1.4 | $120$ | $2$ | $2$ | $1$ |
| 280.48.0-28.a.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-28.b.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.192.5-28.a.1.8 | $280$ | $8$ | $8$ | $5$ |
| 280.504.16-28.a.1.13 | $280$ | $21$ | $21$ | $16$ |
| 40.48.0-40.a.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.c.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.j.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.k.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.1-40.a.1.6 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1-40.b.1.8 | $40$ | $2$ | $2$ | $1$ |
| 280.48.0-56.a.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.c.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.j.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.k.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.1-56.a.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-56.b.1.4 | $280$ | $2$ | $2$ | $1$ |
| 120.48.0-60.a.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.b.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.a.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.c.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.v.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.w.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1-120.a.1.14 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.b.1.5 | $120$ | $2$ | $2$ | $1$ |
| 280.48.0-140.a.1.12 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.b.1.8 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.a.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.c.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.v.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.w.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.1-280.a.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.b.1.4 | $280$ | $2$ | $2$ | $1$ |