Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| Index: | $1920$ | $\PSL_2$-index: | $960$ | ||||
| Genus: | $61 = 1 + \frac{ 960 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
| Cusps: | $40$ (none of which are rational) | Cusp widths | $20^{32}\cdot40^{8}$ | Cusp orbits | $4^{4}\cdot8^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $5$ | ||||||
| $\Q$-gonality: | $11 \le \gamma \le 20$ | ||||||
| $\overline{\Q}$-gonality: | $11 \le \gamma \le 20$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.1920.61.2 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&2\\16&9\end{bmatrix}$, $\begin{bmatrix}11&6\\4&5\end{bmatrix}$, $\begin{bmatrix}15&6\\24&9\end{bmatrix}$, $\begin{bmatrix}15&14\\36&21\end{bmatrix}$, $\begin{bmatrix}25&12\\28&13\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $C_{12}.C_2^5$ |
| Contains $-I$: | no $\quad$ (see 40.960.61.bf.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $48$ |
| Full 40-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{172}\cdot5^{122}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{29}\cdot2^{8}\cdot4^{4}$ |
| Newforms: | 50.2.a.b$^{8}$, 100.2.a.a$^{3}$, 200.2.a.b$^{2}$, 200.2.a.c$^{2}$, 200.2.a.d$^{2}$, 200.2.a.e$^{4}$, 200.2.d.a$^{4}$, 200.2.d.c$^{4}$, 200.2.d.e$^{2}$, 200.2.d.f$^{2}$, 400.2.a.a$^{2}$, 400.2.a.b, 400.2.a.c, 400.2.a.e, 400.2.a.f$^{2}$, 400.2.a.g |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=3,7,\ldots,263$, 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 |
|---|---|---|---|---|---|---|
| 5.20.0.b.1 | $5$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.c.1.1 | $8$ | $20$ | $20$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.960.29-20.f.1.1 | $20$ | $2$ | $2$ | $29$ | $5$ | $2^{8}\cdot4^{4}$ |
| 40.480.16-40.f.1.1 | $40$ | $4$ | $4$ | $16$ | $1$ | $1^{21}\cdot2^{4}\cdot4^{4}$ |
| 40.960.29-20.f.1.18 | $40$ | $2$ | $2$ | $29$ | $5$ | $2^{8}\cdot4^{4}$ |
| 40.960.29-40.ep.1.4 | $40$ | $2$ | $2$ | $29$ | $1$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.960.29-40.ep.1.28 | $40$ | $2$ | $2$ | $29$ | $1$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.960.29-40.ep.2.3 | $40$ | $2$ | $2$ | $29$ | $1$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.960.29-40.ep.2.25 | $40$ | $2$ | $2$ | $29$ | $1$ | $1^{16}\cdot2^{4}\cdot4^{2}$ |
| 40.960.31-40.d.1.1 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.960.31-40.d.1.16 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.960.31-40.l.1.4 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.960.31-40.l.1.13 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.960.31-40.dl.1.3 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.960.31-40.dl.1.14 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.960.31-40.dl.2.8 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
| 40.960.31-40.dl.2.9 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{14}\cdot2^{4}\cdot4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.3840.129-40.ft.1.1 | $40$ | $2$ | $2$ | $129$ | $25$ | $1^{26}\cdot2^{13}\cdot4^{4}$ |
| 40.3840.129-40.ft.2.1 | $40$ | $2$ | $2$ | $129$ | $25$ | $1^{26}\cdot2^{13}\cdot4^{4}$ |
| 40.3840.129-40.fu.1.1 | $40$ | $2$ | $2$ | $129$ | $19$ | $1^{26}\cdot2^{13}\cdot4^{4}$ |
| 40.3840.129-40.fu.2.1 | $40$ | $2$ | $2$ | $129$ | $19$ | $1^{26}\cdot2^{13}\cdot4^{4}$ |
| 40.3840.129-40.fv.1.1 | $40$ | $2$ | $2$ | $129$ | $15$ | $1^{26}\cdot2^{13}\cdot4^{4}$ |
| 40.3840.129-40.fv.2.3 | $40$ | $2$ | $2$ | $129$ | $15$ | $1^{26}\cdot2^{13}\cdot4^{4}$ |
| 40.3840.129-40.fw.1.1 | $40$ | $2$ | $2$ | $129$ | $20$ | $1^{26}\cdot2^{13}\cdot4^{4}$ |
| 40.3840.129-40.fw.2.5 | $40$ | $2$ | $2$ | $129$ | $20$ | $1^{26}\cdot2^{13}\cdot4^{4}$ |
| 40.3840.137-40.gh.1.4 | $40$ | $2$ | $2$ | $137$ | $21$ | $1^{26}\cdot2^{7}\cdot4^{7}\cdot8$ |
| 40.3840.137-40.gi.1.4 | $40$ | $2$ | $2$ | $137$ | $16$ | $1^{26}\cdot2^{7}\cdot4^{7}\cdot8$ |
| 40.3840.137-40.gj.1.4 | $40$ | $2$ | $2$ | $137$ | $19$ | $1^{26}\cdot2^{7}\cdot4^{7}\cdot8$ |
| 40.3840.137-40.gk.1.4 | $40$ | $2$ | $2$ | $137$ | $15$ | $1^{26}\cdot2^{7}\cdot4^{7}\cdot8$ |
| 40.5760.181-40.bz.1.1 | $40$ | $3$ | $3$ | $181$ | $13$ | $1^{56}\cdot2^{8}\cdot4^{12}$ |