Invariants
Level: | $40$ | $\SL_2$-level: | $4$ | ||||
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: | none |
Other labels
Cummins and Pauli (CP) label: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.0.112 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&12\\8&35\end{bmatrix}$, $\begin{bmatrix}11&20\\24&31\end{bmatrix}$, $\begin{bmatrix}15&38\\16&7\end{bmatrix}$, $\begin{bmatrix}25&4\\14&15\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 20.12.0.b.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 631 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8\cdot5^2}\cdot\frac{x^{12}(25x^{4}-1280x^{2}y^{2}+65536y^{4})^{3}}{y^{4}x^{16}(5x^{2}-256y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.12.0-2.a.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
40.12.0-2.a.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
40.48.0-20.b.1.1 | $40$ | $2$ | $2$ | $0$ |
40.48.0-20.b.1.4 | $40$ | $2$ | $2$ | $0$ |
40.48.0-20.c.1.5 | $40$ | $2$ | $2$ | $0$ |
40.48.0-20.c.1.7 | $40$ | $2$ | $2$ | $0$ |
40.48.0-40.d.1.1 | $40$ | $2$ | $2$ | $0$ |
40.48.0-40.d.1.8 | $40$ | $2$ | $2$ | $0$ |
40.48.0-40.g.1.4 | $40$ | $2$ | $2$ | $0$ |
40.48.0-40.g.1.5 | $40$ | $2$ | $2$ | $0$ |
40.120.4-20.d.1.3 | $40$ | $5$ | $5$ | $4$ |
40.144.3-20.d.1.4 | $40$ | $6$ | $6$ | $3$ |
40.240.7-20.d.1.5 | $40$ | $10$ | $10$ | $7$ |
120.48.0-60.e.1.1 | $120$ | $2$ | $2$ | $0$ |
120.48.0-60.e.1.2 | $120$ | $2$ | $2$ | $0$ |
120.48.0-60.g.1.1 | $120$ | $2$ | $2$ | $0$ |
120.48.0-60.g.1.3 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.l.1.5 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.l.1.6 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.r.1.5 | $120$ | $2$ | $2$ | $0$ |
120.48.0-120.r.1.6 | $120$ | $2$ | $2$ | $0$ |
120.72.2-60.b.1.15 | $120$ | $3$ | $3$ | $2$ |
120.96.1-60.b.1.20 | $120$ | $4$ | $4$ | $1$ |
280.48.0-140.f.1.5 | $280$ | $2$ | $2$ | $0$ |
280.48.0-140.f.1.7 | $280$ | $2$ | $2$ | $0$ |
280.48.0-140.g.1.5 | $280$ | $2$ | $2$ | $0$ |
280.48.0-140.g.1.6 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.n.1.5 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.n.1.6 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.q.1.5 | $280$ | $2$ | $2$ | $0$ |
280.48.0-280.q.1.6 | $280$ | $2$ | $2$ | $0$ |
280.192.5-140.b.1.1 | $280$ | $8$ | $8$ | $5$ |
280.504.16-140.b.1.2 | $280$ | $21$ | $21$ | $16$ |