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 | $1^{2}\cdot2\cdot8$ | 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 $\quad(D =$ $-16$) |
Other labels
| Cummins and Pauli (CP) label: | 8C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.0.202 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&28\\15&23\end{bmatrix}$, $\begin{bmatrix}11&12\\5&1\end{bmatrix}$, $\begin{bmatrix}31&20\\25&27\end{bmatrix}$, $\begin{bmatrix}39&16\\16&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.o.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| 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 1491 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}+16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x^{2}+16y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 20.12.0-4.c.1.2 | $20$ | $2$ | $2$ | $0$ | $0$ |
| 40.12.0-4.c.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 |
|---|---|---|---|---|
| 40.48.0-8.i.1.10 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.j.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.w.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.y.1.2 | $40$ | $2$ | $2$ | $0$ |
| 80.48.0-16.i.1.6 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-16.j.1.6 | $80$ | $2$ | $2$ | $0$ |
| 80.48.1-16.c.1.3 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1-16.d.1.3 | $80$ | $2$ | $2$ | $1$ |
| 120.48.0-24.bo.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.bq.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.bs.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.bu.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.72.2-24.cw.1.14 | $120$ | $3$ | $3$ | $2$ |
| 120.96.1-24.iw.1.13 | $120$ | $4$ | $4$ | $1$ |
| 40.48.0-40.bs.1.6 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.bu.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.bw.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.by.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.120.4-40.bq.1.13 | $40$ | $5$ | $5$ | $4$ |
| 40.144.3-40.cg.1.25 | $40$ | $6$ | $6$ | $3$ |
| 40.240.7-40.cw.1.21 | $40$ | $10$ | $10$ | $7$ |
| 240.48.0-48.i.1.12 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-48.j.1.10 | $240$ | $2$ | $2$ | $0$ |
| 240.48.1-48.c.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-48.d.1.5 | $240$ | $2$ | $2$ | $1$ |
| 280.48.0-56.bm.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bo.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bq.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bs.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.192.5-56.bq.1.29 | $280$ | $8$ | $8$ | $5$ |
| 280.504.16-56.cw.1.2 | $280$ | $21$ | $21$ | $16$ |
| 80.48.0-80.q.1.16 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.r.1.15 | $80$ | $2$ | $2$ | $0$ |
| 80.48.1-80.c.1.2 | $80$ | $2$ | $2$ | $1$ |
| 80.48.1-80.d.1.1 | $80$ | $2$ | $2$ | $1$ |
| 120.48.0-120.dw.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dy.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ee.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.eg.1.6 | $120$ | $2$ | $2$ | $0$ |
| 240.48.0-240.q.1.29 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.r.1.25 | $240$ | $2$ | $2$ | $0$ |
| 240.48.1-240.c.1.6 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-240.d.1.2 | $240$ | $2$ | $2$ | $1$ |
| 280.48.0-280.dw.1.15 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.dy.1.10 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.ee.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.eg.1.6 | $280$ | $2$ | $2$ | $0$ |