Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $800$ | ||
| 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 | $10^{8}\cdot20^{4}\cdot40^{8}$ | Cusp orbits | $2^{2}\cdot4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 10$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.960.31.1580 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&8\\0&29\end{bmatrix}$, $\begin{bmatrix}11&8\\14&13\end{bmatrix}$, $\begin{bmatrix}15&12\\16&3\end{bmatrix}$, $\begin{bmatrix}15&36\\12&35\end{bmatrix}$, $\begin{bmatrix}29&16\\18&21\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_4\times C_8:D_6$ |
| Contains $-I$: | no $\quad$ (see 40.480.31.er.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $48$ |
| Full 40-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{102}\cdot5^{54}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2^{4}\cdot4^{2}$ |
| Newforms: | 20.2.a.a, 40.2.a.a, 50.2.a.a, 50.2.a.b$^{3}$, 80.2.a.a, 80.2.a.b, 100.2.a.a$^{2}$, 160.2.d.a, 200.2.a.a, 200.2.a.c, 200.2.a.e, 200.2.d.a, 200.2.d.c, 200.2.d.f, 400.2.a.d, 400.2.a.h, 800.2.d.b, 800.2.d.d |
Rational points
This modular curve has no $\Q_p$ points for $p=3,7$, 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.w.1.1 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 40.480.15-40.t.2.1 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.t.2.38 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.z.2.5 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.z.2.34 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.cd.1.1 | $40$ | $2$ | $2$ | $15$ | $1$ | $2^{4}\cdot4^{2}$ |
| 40.480.15-40.cd.1.25 | $40$ | $2$ | $2$ | $15$ | $1$ | $2^{4}\cdot4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1920.61-40.he.2.1 | $40$ | $2$ | $2$ | $61$ | $6$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.hi.2.7 | $40$ | $2$ | $2$ | $61$ | $11$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.hu.2.2 | $40$ | $2$ | $2$ | $61$ | $4$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.hy.2.1 | $40$ | $2$ | $2$ | $61$ | $9$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.ik.2.1 | $40$ | $2$ | $2$ | $61$ | $6$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.io.2.5 | $40$ | $2$ | $2$ | $61$ | $9$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.ja.2.1 | $40$ | $2$ | $2$ | $61$ | $6$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.je.2.3 | $40$ | $2$ | $2$ | $61$ | $7$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.jm.1.1 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.jn.1.1 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.nr.1.5 | $40$ | $2$ | $2$ | $65$ | $6$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.nt.1.1 | $40$ | $2$ | $2$ | $65$ | $5$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.re.1.1 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.rf.1.1 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.rq.1.3 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.rr.1.1 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bks.1.1 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bkt.1.2 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bkw.1.1 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bkx.1.1 | $40$ | $2$ | $2$ | $65$ | $11$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bli.1.1 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.blj.1.3 | $40$ | $2$ | $2$ | $65$ | $7$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.blm.1.1 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bln.1.1 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.2880.91-40.ld.2.11 | $40$ | $3$ | $3$ | $91$ | $5$ | $1^{28}\cdot2^{4}\cdot4^{6}$ |