Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| 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 | $2^{4}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-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: | 40.960.31.1355 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&20\\16&3\end{bmatrix}$, $\begin{bmatrix}7&20\\16&13\end{bmatrix}$, $\begin{bmatrix}9&0\\32&11\end{bmatrix}$, $\begin{bmatrix}11&32\\2&7\end{bmatrix}$, $\begin{bmatrix}27&8\\18&11\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | Group 768.1032997 |
| Contains $-I$: | no $\quad$ (see 40.480.31.bj.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: | $2^{86}\cdot5^{54}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2^{4}\cdot4^{2}$ |
| Newforms: | 20.2.a.a, 40.2.a.a, 40.2.d.a, 50.2.a.a, 50.2.a.b$^{3}$, 80.2.a.a, 80.2.a.b, 100.2.a.a$^{2}$, 200.2.a.a, 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, 400.2.a.d, 400.2.a.h |
Rational points
This modular curve has no $\Q_p$ points for $p=3$, 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 |
|---|---|---|---|---|---|---|
| 40.96.0-40.l.2.14 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 40.480.15-20.e.1.20 | $40$ | $2$ | $2$ | $15$ | $1$ | $2^{4}\cdot4^{2}$ |
| 40.480.15-20.e.1.22 | $40$ | $2$ | $2$ | $15$ | $1$ | $2^{4}\cdot4^{2}$ |
| 40.480.15-40.y.1.27 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.y.1.41 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.z.2.23 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.z.2.38 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
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.dz.2.13 | $40$ | $2$ | $2$ | $61$ | $4$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.ed.2.11 | $40$ | $2$ | $2$ | $61$ | $13$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.et.1.10 | $40$ | $2$ | $2$ | $61$ | $8$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.ex.2.10 | $40$ | $2$ | $2$ | $61$ | $5$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.fn.2.9 | $40$ | $2$ | $2$ | $61$ | $9$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.fr.2.9 | $40$ | $2$ | $2$ | $61$ | $4$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.gh.2.12 | $40$ | $2$ | $2$ | $61$ | $9$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.gl.2.15 | $40$ | $2$ | $2$ | $61$ | $6$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.ne.2.14 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.nt.1.9 | $40$ | $2$ | $2$ | $65$ | $5$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.om.1.6 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.oo.2.5 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.ya.2.9 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.yc.1.16 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.yq.1.12 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.ys.2.10 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bdo.1.11 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bdq.2.14 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bee.2.10 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.beg.1.16 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bgq.1.2 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bgs.2.7 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bha.2.14 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bhb.1.1 | $40$ | $2$ | $2$ | $65$ | $6$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.2880.91-40.if.2.16 | $40$ | $3$ | $3$ | $91$ | $5$ | $1^{28}\cdot2^{4}\cdot4^{6}$ |