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 | $10^{8}\cdot20^{4}\cdot40^{8}$ | Cusp orbits | $2^{4}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-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: | 40.960.31.30 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&16\\2&3\end{bmatrix}$, $\begin{bmatrix}11&16\\18&19\end{bmatrix}$, $\begin{bmatrix}31&4\\28&21\end{bmatrix}$, $\begin{bmatrix}33&32\\16&37\end{bmatrix}$, $\begin{bmatrix}39&12\\4&29\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_4\times C_8:D_6$ |
| Contains $-I$: | no $\quad$ (see 40.480.31.fr.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{86}\cdot5^{62}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2^{4}\cdot4^{2}$ |
| Newforms: | 50.2.a.b$^{4}$, 100.2.a.a$^{3}$, 200.2.a.c$^{2}$, 200.2.a.e$^{2}$, 200.2.d.a$^{2}$, 200.2.d.c$^{2}$, 200.2.d.f$^{2}$, 400.2.a.a, 400.2.a.c, 400.2.a.e, 400.2.a.f |
Rational points
This modular curve has no $\Q_p$ points for $p=3,7,17$, 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.l.1.4 | $8$ | $10$ | $10$ | $0$ | $0$ | 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.l.1.4 | $8$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 40.480.15-40.z.2.23 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.z.2.32 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.z.2.56 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.z.2.63 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.480.15-40.ch.1.23 | $40$ | $2$ | $2$ | $15$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.480.15-40.ch.1.48 | $40$ | $2$ | $2$ | $15$ | $2$ | $2^{4}\cdot4^{2}$ |
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.jx.2.12 | $40$ | $2$ | $2$ | $61$ | $5$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.kd.2.12 | $40$ | $2$ | $2$ | $61$ | $12$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.kv.2.9 | $40$ | $2$ | $2$ | $61$ | $7$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.lb.2.10 | $40$ | $2$ | $2$ | $61$ | $6$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.lt.2.9 | $40$ | $2$ | $2$ | $61$ | $5$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.lz.2.11 | $40$ | $2$ | $2$ | $61$ | $12$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.mr.2.12 | $40$ | $2$ | $2$ | $61$ | $9$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.61-40.mx.2.10 | $40$ | $2$ | $2$ | $61$ | $10$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.1920.65-40.ns.1.5 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.ns.1.14 | $40$ | $2$ | $2$ | $65$ | $9$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.rz.1.7 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.rz.1.11 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bhb.1.1 | $40$ | $2$ | $2$ | $65$ | $6$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bhb.1.10 | $40$ | $2$ | $2$ | $65$ | $6$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bhj.1.5 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bhj.1.9 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{14}\cdot2^{6}\cdot4^{2}$ |
| 40.1920.65-40.bmp.1.11 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bmp.1.13 | $40$ | $2$ | $2$ | $65$ | $12$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bmx.1.9 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bmx.1.14 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bnv.1.12 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bnv.1.14 | $40$ | $2$ | $2$ | $65$ | $10$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bod.1.11 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.1920.65-40.bod.1.16 | $40$ | $2$ | $2$ | $65$ | $8$ | $1^{12}\cdot2^{7}\cdot4^{2}$ |
| 40.2880.91-40.qz.2.18 | $40$ | $3$ | $3$ | $91$ | $6$ | $1^{28}\cdot2^{4}\cdot4^{6}$ |
