Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $576$ | $\PSL_2$-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 | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $5$ | ||||||
| $\Q$-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: | 40.576.17.3810 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}17&1\\28&35\end{bmatrix}$, $\begin{bmatrix}19&0\\6&33\end{bmatrix}$, $\begin{bmatrix}21&12\\14&39\end{bmatrix}$, $\begin{bmatrix}31&11\\24&33\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $(C_2\times D_{20}):C_4^2$ |
| Contains $-I$: | no $\quad$ (see 40.288.17.ji.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $1280$ |
Jacobian
| Conductor: | $2^{84}\cdot5^{27}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2$ |
| Newforms: | 20.2.a.a, 40.2.a.a, 80.2.a.a, 100.2.a.a, 200.2.a.c, 320.2.a.a$^{2}$, 320.2.a.d, 320.2.a.f, 800.2.a.a, 800.2.a.i, 1600.2.a.bc, 1600.2.a.c, 1600.2.a.n$^{2}$, 1600.2.a.o |
Rational points
This modular curve has no $\Q_p$ points for $p=7,13,53$, 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 |
|---|---|---|---|---|---|---|
| 40.96.1-40.cg.1.1 | $40$ | $6$ | $6$ | $1$ | $0$ | $1^{14}\cdot2$ |
| 40.288.7-40.cc.1.6 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{8}\cdot2$ |
| 40.288.7-40.cc.1.9 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{8}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1152.33-40.beg.1.5 | $40$ | $2$ | $2$ | $33$ | $5$ | $2^{6}\cdot4$ |
| 40.1152.33-40.beg.2.5 | $40$ | $2$ | $2$ | $33$ | $5$ | $2^{6}\cdot4$ |
| 40.1152.33-40.bek.1.1 | $40$ | $2$ | $2$ | $33$ | $5$ | $2^{6}\cdot4$ |
| 40.1152.33-40.bek.2.1 | $40$ | $2$ | $2$ | $33$ | $5$ | $2^{6}\cdot4$ |
| 40.1152.33-40.bfa.1.1 | $40$ | $2$ | $2$ | $33$ | $5$ | $2^{6}\cdot4$ |
| 40.1152.33-40.bfa.2.1 | $40$ | $2$ | $2$ | $33$ | $5$ | $2^{6}\cdot4$ |
| 40.1152.33-40.bfe.1.1 | $40$ | $2$ | $2$ | $33$ | $5$ | $2^{6}\cdot4$ |
| 40.1152.33-40.bfe.2.1 | $40$ | $2$ | $2$ | $33$ | $5$ | $2^{6}\cdot4$ |
| 40.2880.97-40.cfl.1.3 | $40$ | $5$ | $5$ | $97$ | $33$ | $1^{68}\cdot2^{6}$ |