Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $800$ | ||
| Index: | $960$ | $\PSL_2$-index: | $480$ | ||||
| Genus: | $33 = 1 + \frac{ 480 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $20^{8}\cdot40^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.960.33.5696 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&28\\12&23\end{bmatrix}$, $\begin{bmatrix}9&0\\30&19\end{bmatrix}$, $\begin{bmatrix}17&24\\6&3\end{bmatrix}$, $\begin{bmatrix}33&20\\30&33\end{bmatrix}$, $\begin{bmatrix}39&8\\34&31\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_4\times C_8:D_6$ |
| Contains $-I$: | no $\quad$ (see 40.480.33.nb.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{111}\cdot5^{66}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{7}\cdot4^{2}$ |
| Newforms: | 50.2.a.b$^{3}$, 100.2.a.a$^{2}$, 200.2.a.c, 200.2.a.e, 200.2.d.a, 200.2.d.b, 200.2.d.c, 200.2.d.d, 200.2.d.e, 200.2.d.f, 800.2.a.b, 800.2.a.e, 800.2.a.f, 800.2.a.h, 800.2.a.j, 800.2.a.l, 800.2.a.n |
Rational points
This modular curve has no $\Q_p$ points for $p=3,13$, 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 |
|---|---|---|---|---|---|---|
| 40.480.15-40.u.2.9 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.480.15-40.u.2.27 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.480.15-40.z.2.3 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.480.15-40.z.2.23 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.480.17-40.db.1.13 | $40$ | $2$ | $2$ | $17$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.480.17-40.db.1.29 | $40$ | $2$ | $2$ | $17$ | $4$ | $2^{4}\cdot4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1920.65-40.ey.2.2 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.ge.2.1 | $40$ | $2$ | $2$ | $65$ | $16$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.ka.2.11 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.nn.2.11 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.qr.2.13 | $40$ | $2$ | $2$ | $65$ | $14$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.rb.2.10 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.vr.2.2 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.ws.1.1 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bct.1.3 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bda.2.2 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bdy.2.13 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bee.2.10 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bff.2.9 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bfl.2.10 | $40$ | $2$ | $2$ | $65$ | $14$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bfu.2.5 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bgd.2.2 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bkc.2.13 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bki.2.10 | $40$ | $2$ | $2$ | $65$ | $14$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bli.1.13 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.blo.1.13 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bnc.2.13 | $40$ | $2$ | $2$ | $65$ | $14$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bnm.1.13 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bny.2.11 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bod.1.11 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.2880.97-40.bsr.2.29 | $40$ | $3$ | $3$ | $97$ | $12$ | $1^{22}\cdot2^{9}\cdot4^{6}$ |