Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| 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: | $6$ | ||||||
| $\Q$-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: | 40.960.33.123 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&4\\18&9\end{bmatrix}$, $\begin{bmatrix}9&32\\28&17\end{bmatrix}$, $\begin{bmatrix}17&24\\22&3\end{bmatrix}$, $\begin{bmatrix}17&28\\22&9\end{bmatrix}$, $\begin{bmatrix}31&8\\32&33\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_4\times C_8:D_6$ |
| Contains $-I$: | no $\quad$ (see 40.480.33.gg.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: | $2^{137}\cdot5^{64}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}\cdot2^{6}\cdot4^{2}$ |
| Newforms: | 50.2.a.b$^{3}$, 64.2.a.a, 100.2.a.a$^{2}$, 200.2.a.c, 200.2.a.e, 200.2.d.a, 200.2.d.c, 200.2.d.f, 800.2.d.a, 800.2.d.c, 800.2.d.e, 1600.2.a.bb, 1600.2.a.bc, 1600.2.a.e, 1600.2.a.h, 1600.2.a.n, 1600.2.a.r, 1600.2.a.t |
Rational points
This modular curve has no $\Q_p$ points for $p=3$, and therefore no rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.m.2.5 | $8$ | $10$ | $10$ | $1$ | $0$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.m.2.5 | $8$ | $10$ | $10$ | $1$ | $0$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 40.480.15-40.s.2.2 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.15-40.s.2.30 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.15-40.z.2.9 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.15-40.z.2.23 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.17-40.bv.1.6 | $40$ | $2$ | $2$ | $17$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.480.17-40.bv.1.26 | $40$ | $2$ | $2$ | $17$ | $4$ | $2^{4}\cdot4^{2}$ |
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.ft.2.5 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.ge.2.1 | $40$ | $2$ | $2$ | $65$ | $16$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.hx.2.3 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.ii.2.3 | $40$ | $2$ | $2$ | $65$ | $14$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.iq.1.2 | $40$ | $2$ | $2$ | $65$ | $16$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.nl.1.11 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.oc.1.9 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.om.1.6 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.ou.2.4 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.pa.1.2 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.pm.1.4 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.po.2.4 | $40$ | $2$ | $2$ | $65$ | $16$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.qc.1.4 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.qe.1.5 | $40$ | $2$ | $2$ | $65$ | $16$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.qq.1.6 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.qw.1.2 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.re.1.7 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.ro.2.6 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.rs.2.2 | $40$ | $2$ | $2$ | $65$ | $16$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.rz.1.7 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.ub.2.6 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.um.2.7 | $40$ | $2$ | $2$ | $65$ | $16$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.wf.2.1 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.wq.2.3 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.2880.97-40.kv.2.4 | $40$ | $3$ | $3$ | $97$ | $15$ | $1^{22}\cdot2^{9}\cdot4^{6}$ |