Invariants
Level: | $40$ | $\SL_2$-level: | $4$ | ||||
Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
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: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.12.0.53 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&34\\24&35\end{bmatrix}$, $\begin{bmatrix}19&22\\25&37\end{bmatrix}$, $\begin{bmatrix}33&14\\31&17\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 40-isogeny field degree: | $24$ |
Cyclic 40-torsion field degree: | $384$ |
Full 40-torsion field degree: | $61440$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ x^{2} - 3 x z + 160 y^{2} + z^{2} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.6.0.f.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
10.6.0.a.1 | $10$ | $2$ | $2$ | $0$ | $0$ |
40.6.0.c.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 |
---|---|---|---|---|
40.60.4.by.1 | $40$ | $5$ | $5$ | $4$ |
40.72.3.dy.1 | $40$ | $6$ | $6$ | $3$ |
40.120.7.fy.1 | $40$ | $10$ | $10$ | $7$ |
120.36.2.qq.1 | $120$ | $3$ | $3$ | $2$ |
120.48.1.bzo.1 | $120$ | $4$ | $4$ | $1$ |
280.96.5.fe.1 | $280$ | $8$ | $8$ | $5$ |
280.252.16.kc.1 | $280$ | $21$ | $21$ | $16$ |
280.336.21.kc.1 | $280$ | $28$ | $28$ | $21$ |