Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $960$ | $\PSL_2$-index: | $480$ | ||||
| Genus: | $31 = 1 + \frac{ 480 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $20^{16}\cdot40^{4}$ | Cusp orbits | $2^{2}\cdot4^{2}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $6$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 10$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 10$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.960.31.148 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&0\\10&31\end{bmatrix}$, $\begin{bmatrix}9&16\\14&23\end{bmatrix}$, $\begin{bmatrix}9&20\\30&29\end{bmatrix}$, $\begin{bmatrix}23&16\\38&27\end{bmatrix}$, $\begin{bmatrix}39&8\\12&21\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_4\times C_8:D_6$ |
| Contains $-I$: | no $\quad$ (see 40.480.31.t.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: | $2^{125}\cdot5^{62}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2^{4}\cdot4^{2}$ |
| 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.c, 200.2.d.f, 800.2.d.a, 800.2.d.c, 800.2.d.e, 1600.2.a.a, 1600.2.a.c, 1600.2.a.i, 1600.2.a.k, 1600.2.a.o, 1600.2.a.q, 1600.2.a.w, 1600.2.a.y |
Rational points
This modular curve has no $\Q_p$ points for $p=3,17$, 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 |
| 8.96.0-8.e.1.8 | $8$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.e.1.8 | $8$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 40.480.15-40.c.1.13 | $40$ | $2$ | $2$ | $15$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.480.15-40.c.1.30 | $40$ | $2$ | $2$ | $15$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.480.15-40.s.2.31 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.s.2.35 | $40$ | $2$ | $2$ | $15$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.z.2.23 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.z.2.33 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1920.61-40.p.2.16 | $40$ | $2$ | $2$ | $61$ | $11$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.v.2.11 | $40$ | $2$ | $2$ | $61$ | $14$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.bn.2.10 | $40$ | $2$ | $2$ | $61$ | $13$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.bt.2.9 | $40$ | $2$ | $2$ | $61$ | $12$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.cl.2.16 | $40$ | $2$ | $2$ | $61$ | $13$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.cr.2.12 | $40$ | $2$ | $2$ | $61$ | $14$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.dj.1.15 | $40$ | $2$ | $2$ | $61$ | $13$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.dp.2.10 | $40$ | $2$ | $2$ | $61$ | $8$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.je.1.12 | $40$ | $2$ | $2$ | $65$ | $17$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.nz.2.9 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.od.2.1 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.on.1.14 | $40$ | $2$ | $2$ | $65$ | $13$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.ov.2.14 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.pb.2.10 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.pn.2.14 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.pp.2.14 | $40$ | $2$ | $2$ | $65$ | $16$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.qd.2.16 | $40$ | $2$ | $2$ | $65$ | $14$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.qf.2.9 | $40$ | $2$ | $2$ | $65$ | $18$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.qr.2.13 | $40$ | $2$ | $2$ | $65$ | $14$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.qx.2.16 | $40$ | $2$ | $2$ | $65$ | $14$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.rf.1.10 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.rp.2.10 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.rs.2.2 | $40$ | $2$ | $2$ | $65$ | $16$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.rz.1.11 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.2880.91-40.by.2.15 | $40$ | $3$ | $3$ | $91$ | $17$ | $1^{28}\cdot2^{4}\cdot4^{6}$ |