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 | $2^{3}$ | ||
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: | 40.48.0.786 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}19&16\\14&13\end{bmatrix}$, $\begin{bmatrix}21&0\\25&13\end{bmatrix}$, $\begin{bmatrix}39&12\\37&39\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.24.0.bx.1 for the level structure with $-I$) |
Cyclic 40-isogeny field degree: | $12$ |
Cyclic 40-torsion field degree: | $96$ |
Full 40-torsion field degree: | $15360$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 5 x^{2} + 16 y^{2} - 10 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
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.24.0-8.p.1.7 | $8$ | $2$ | $2$ | $0$ | $0$ |
40.24.0-20.h.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.24.0-20.h.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.24.0-8.p.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.24.0-40.bb.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.24.0-40.bb.1.13 | $40$ | $2$ | $2$ | $0$ | $0$ |
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.cx.1.3 | $40$ | $5$ | $5$ | $8$ |
40.288.7-40.fc.1.13 | $40$ | $6$ | $6$ | $7$ |
40.480.15-40.gn.1.6 | $40$ | $10$ | $10$ | $15$ |
120.144.4-120.lx.1.9 | $120$ | $3$ | $3$ | $4$ |
120.192.3-120.pf.1.9 | $120$ | $4$ | $4$ | $3$ |
280.384.11-280.il.1.10 | $280$ | $8$ | $8$ | $11$ |