Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $288$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $17 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}\cdot20^{4}\cdot40^{4}$ | Cusp orbits | $4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $7 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $7 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40X17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.288.17.1120 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}19&16\\35&1\end{bmatrix}$, $\begin{bmatrix}19&22\\25&37\end{bmatrix}$, $\begin{bmatrix}19&34\\15&1\end{bmatrix}$, $\begin{bmatrix}23&10\\35&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $64$ |
| Full 40-torsion field degree: | $2560$ |
Jacobian
| Conductor: | $2^{84}\cdot5^{27}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2$ |
| Newforms: | 80.2.a.a, 100.2.a.a$^{2}$, 160.2.a.a, 160.2.a.b, 200.2.a.c$^{2}$, 320.2.a.a, 320.2.a.c, 320.2.a.d, 320.2.a.f, 1600.2.a.bc, 1600.2.a.c, 1600.2.a.n$^{2}$, 1600.2.a.w |
Rational points
This modular curve has no $\Q_p$ points for $p=3,7,43$, 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 |
|---|---|---|---|---|---|---|
| 40.48.1.ic.1 | $40$ | $6$ | $6$ | $1$ | $0$ | $1^{14}\cdot2$ |
| 40.144.7.hf.1 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{8}\cdot2$ |
| 40.144.7.ig.1 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{8}\cdot2$ |
| 40.144.7.pm.1 | $40$ | $2$ | $2$ | $7$ | $3$ | $1^{8}\cdot2$ |
| 40.144.7.qd.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{8}\cdot2$ |
| 40.144.9.hl.1 | $40$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 40.144.9.hu.1 | $40$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 40.144.9.kj.1 | $40$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.576.33.cqw.1 | $40$ | $2$ | $2$ | $33$ | $4$ | $2^{6}\cdot4$ |
| 40.576.33.cqw.2 | $40$ | $2$ | $2$ | $33$ | $4$ | $2^{6}\cdot4$ |
| 40.576.33.cra.1 | $40$ | $2$ | $2$ | $33$ | $4$ | $2^{6}\cdot4$ |
| 40.576.33.cra.2 | $40$ | $2$ | $2$ | $33$ | $4$ | $2^{6}\cdot4$ |
| 40.576.33.csg.1 | $40$ | $2$ | $2$ | $33$ | $4$ | $2^{6}\cdot4$ |
| 40.576.33.csg.2 | $40$ | $2$ | $2$ | $33$ | $4$ | $2^{6}\cdot4$ |
| 40.576.33.csk.1 | $40$ | $2$ | $2$ | $33$ | $4$ | $2^{6}\cdot4$ |
| 40.576.33.csk.2 | $40$ | $2$ | $2$ | $33$ | $4$ | $2^{6}\cdot4$ |
| 40.1440.97.hju.1 | $40$ | $5$ | $5$ | $97$ | $32$ | $1^{68}\cdot2^{6}$ |