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 | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-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: | 40.960.33.1716 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&28\\20&19\end{bmatrix}$, $\begin{bmatrix}3&16\\26&11\end{bmatrix}$, $\begin{bmatrix}9&12\\30&1\end{bmatrix}$, $\begin{bmatrix}11&36\\6&29\end{bmatrix}$, $\begin{bmatrix}27&32\\14&3\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_4\times C_8:D_6$ |
| Contains $-I$: | no $\quad$ (see 40.480.33.iz.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{111}\cdot5^{56}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}\cdot2^{6}\cdot4^{2}$ |
| Newforms: | 32.2.a.a, 40.2.d.a, 50.2.a.b$^{3}$, 100.2.a.a$^{2}$, 160.2.a.a, 160.2.a.b, 160.2.a.c, 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.f, 800.2.a.b, 800.2.a.d, 800.2.a.h, 800.2.a.l |
Rational points
This modular curve has no $\Q_p$ points for $p=3,17$, 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.bf.2.4 | $40$ | $10$ | $10$ | $1$ | $1$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 40.480.15-40.y.1.10 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.15-40.y.1.20 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.15-40.z.2.14 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.15-40.z.2.23 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.17-40.cg.1.8 | $40$ | $2$ | $2$ | $17$ | $3$ | $2^{4}\cdot4^{2}$ |
| 40.480.17-40.cg.1.24 | $40$ | $2$ | $2$ | $17$ | $3$ | $2^{4}\cdot4^{2}$ |
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.nd.2.14 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.nr.1.14 | $40$ | $2$ | $2$ | $65$ | $6$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.on.1.14 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.op.1.13 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.xb.2.6 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.xf.2.7 | $40$ | $2$ | $2$ | $65$ | $15$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.yb.2.6 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.yd.1.8 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.yr.1.8 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.yt.2.4 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bad.2.5 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bah.2.8 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bcp.2.7 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bct.1.3 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bdp.1.7 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bdr.2.8 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bef.2.6 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.beh.1.8 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bfr.2.8 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bfv.2.8 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bgr.1.10 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bgt.1.15 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bha.2.14 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.bhb.1.10 | $40$ | $2$ | $2$ | $65$ | $6$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.2880.97-40.bki.2.3 | $40$ | $3$ | $3$ | $97$ | $11$ | $1^{22}\cdot2^{9}\cdot4^{6}$ |