Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $240$ | $\PSL_2$-index: | $240$ | ||||
| Genus: | $17 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $20^{4}\cdot40^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40A17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.17.363 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&15\\32&31\end{bmatrix}$, $\begin{bmatrix}13&20\\4&17\end{bmatrix}$, $\begin{bmatrix}33&23\\16&27\end{bmatrix}$, $\begin{bmatrix}35&28\\28&31\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 40.480.17-40.hw.1.1, 40.480.17-40.hw.1.2, 40.480.17-40.hw.1.3, 40.480.17-40.hw.1.4, 80.480.17-40.hw.1.1, 80.480.17-40.hw.1.2, 80.480.17-40.hw.1.3, 80.480.17-40.hw.1.4, 120.480.17-40.hw.1.1, 120.480.17-40.hw.1.2, 120.480.17-40.hw.1.3, 120.480.17-40.hw.1.4, 240.480.17-40.hw.1.1, 240.480.17-40.hw.1.2, 240.480.17-40.hw.1.3, 240.480.17-40.hw.1.4, 280.480.17-40.hw.1.1, 280.480.17-40.hw.1.2, 280.480.17-40.hw.1.3, 280.480.17-40.hw.1.4 |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{76}\cdot5^{34}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2$ |
| Newforms: | 50.2.a.b$^{3}$, 200.2.a.e, 400.2.a.a$^{2}$, 400.2.a.f$^{2}$, 1600.2.a.ba, 1600.2.a.g, 1600.2.a.j$^{2}$, 1600.2.a.n, 1600.2.a.s, 1600.2.a.x$^{2}$ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.120.8.f.1 | $20$ | $2$ | $2$ | $8$ | $2$ | $1^{7}\cdot2$ |
| 40.48.1.ea.1 | $40$ | $5$ | $5$ | $1$ | $0$ | $1^{14}\cdot2$ |
| 40.120.8.cb.1 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{7}\cdot2$ |
| 40.120.8.fg.1 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{7}\cdot2$ |
| 40.120.8.fh.1 | $40$ | $2$ | $2$ | $8$ | $1$ | $1^{7}\cdot2$ |
| 40.120.9.bb.1 | $40$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 40.120.9.dy.1 | $40$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
| 40.120.9.dz.1 | $40$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.720.49.bjs.1 | $40$ | $3$ | $3$ | $49$ | $14$ | $1^{28}\cdot2^{2}$ |
| 40.960.65.cmy.1 | $40$ | $4$ | $4$ | $65$ | $15$ | $1^{40}\cdot2^{4}$ |