Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $1152$ | $\PSL_2$-index: | $576$ | ||||
| Genus: | $33 = 1 + \frac{ 576 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}\cdot20^{8}\cdot40^{8}$ | Cusp orbits | $4^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $5$ | ||||||
| $\Q$-gonality: | $8 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $8 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.1152.33.34386 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&29\\30&17\end{bmatrix}$, $\begin{bmatrix}17&16\\34&19\end{bmatrix}$, $\begin{bmatrix}29&7\\22&19\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $(C_2\times D_{10}):C_4^2$ |
| Contains $-I$: | no $\quad$ (see 40.576.33.bfa.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $640$ |
Jacobian
| Conductor: | $2^{168}\cdot5^{43}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2^{7}\cdot4$ |
| Newforms: | 20.2.a.a, 40.2.a.a, 40.2.c.a, 80.2.a.a, 80.2.c.a, 100.2.a.a, 160.2.c.a, 200.2.a.c, 320.2.a.a$^{2}$, 320.2.a.d, 320.2.a.f, 320.2.c.b, 320.2.c.c$^{2}$, 320.2.c.d, 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 $p=7,11,13,53,61$, 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 |
|---|---|---|---|---|---|---|
| 40.576.13-40.gx.1.2 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}\cdot2^{4}\cdot4$ |
| 40.576.13-40.gx.1.3 | $40$ | $2$ | $2$ | $13$ | $2$ | $1^{8}\cdot2^{4}\cdot4$ |
| 40.576.17-40.ji.1.1 | $40$ | $2$ | $2$ | $17$ | $5$ | $2^{6}\cdot4$ |
| 40.576.17-40.ji.1.4 | $40$ | $2$ | $2$ | $17$ | $5$ | $2^{6}\cdot4$ |
| 40.576.17-40.ju.1.1 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |
| 40.576.17-40.ju.1.6 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.5760.193-40.cuf.1.3 | $40$ | $5$ | $5$ | $193$ | $35$ | $1^{68}\cdot2^{40}\cdot4^{3}$ |