Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.0.1073 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&20\\0&11\end{bmatrix}$, $\begin{bmatrix}9&33\\28&9\end{bmatrix}$, $\begin{bmatrix}35&11\\32&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.be.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $48$ |
| Full 40-torsion field degree: | $7680$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 8 x^{2} - 10 y^{2} + 5 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 |
|---|---|---|---|---|---|
| 8.48.0-8.k.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.k.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.ca.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.ca.1.15 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.cb.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.cb.1.15 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.480.16-40.bq.1.7 | $40$ | $5$ | $5$ | $16$ |
| 40.576.15-40.eb.1.16 | $40$ | $6$ | $6$ | $15$ |
| 40.960.31-40.fu.2.6 | $40$ | $10$ | $10$ | $31$ |
| 80.192.1-80.s.2.2 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.t.2.2 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.u.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.x.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.y.1.8 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.bb.1.8 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.bc.1.11 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.bd.1.7 | $80$ | $2$ | $2$ | $1$ |
| 120.288.8-120.px.2.14 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-120.ke.1.23 | $120$ | $4$ | $4$ | $7$ |
| 240.192.1-240.cf.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.cg.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.cn.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.cs.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.cv.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.da.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.dh.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.di.2.13 | $240$ | $2$ | $2$ | $1$ |