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: | $5$ | ||||||
| $\Q$-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: | 40.960.33.125 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&12\\18&25\end{bmatrix}$, $\begin{bmatrix}25&16\\16&35\end{bmatrix}$, $\begin{bmatrix}31&12\\6&19\end{bmatrix}$, $\begin{bmatrix}39&16\\32&1\end{bmatrix}$, $\begin{bmatrix}39&24\\20&11\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_4\times C_8:D_6$ |
| Contains $-I$: | no $\quad$ (see 40.480.33.fh.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: | $2^{127}\cdot5^{64}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}\cdot2^{6}\cdot4^{2}$ |
| Newforms: | 32.2.a.a, 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.c, 200.2.d.f, 800.2.a.a, 800.2.a.c, 800.2.a.d, 800.2.a.g, 800.2.a.i, 800.2.a.k, 800.2.a.m, 800.2.d.a, 800.2.d.c, 800.2.d.e |
Rational points
This modular curve has no $\Q_p$ points for $p=3,13$, 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.i.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.i.2.5 | $8$ | $10$ | $10$ | $1$ | $0$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 40.480.15-40.s.1.2 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.15-40.s.1.32 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.15-40.z.2.9 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.15-40.z.2.24 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.480.17-40.bu.1.2 | $40$ | $2$ | $2$ | $17$ | $3$ | $2^{4}\cdot4^{2}$ |
| 40.480.17-40.bu.1.23 | $40$ | $2$ | $2$ | $17$ | $3$ | $2^{4}\cdot4^{2}$ |
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.eu.2.2 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.ff.2.5 | $40$ | $2$ | $2$ | $65$ | $15$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.gy.2.4 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.hj.2.3 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.ix.2.2 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.je.1.3 | $40$ | $2$ | $2$ | $65$ | $17$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.jn.1.2 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.js.1.11 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.kd.1.5 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.kg.2.5 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.kt.2.6 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.ku.1.2 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.mc.1.6 | $40$ | $2$ | $2$ | $65$ | $17$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.md.1.3 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.mq.1.2 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.mt.1.2 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.ne.2.2 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.nj.1.2 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.ns.1.2 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.nz.1.9 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.tc.2.5 | $40$ | $2$ | $2$ | $65$ | $15$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.tn.1.2 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.vg.2.3 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.vr.2.5 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{10}\cdot2^{7}\cdot4^{2}$ |
| 40.2880.97-40.jp.2.9 | $40$ | $3$ | $3$ | $97$ | $15$ | $1^{22}\cdot2^{9}\cdot4^{6}$ |