Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $960$ | $\PSL_2$-index: | $480$ | ||||
| Genus: | $35 = 1 + \frac{ 480 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $40^{12}$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $3$ | ||||||
| $\Q$-gonality: | $7 \le \gamma \le 10$ | ||||||
| $\overline{\Q}$-gonality: | $7 \le \gamma \le 10$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.960.35.20 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&8\\24&13\end{bmatrix}$, $\begin{bmatrix}9&0\\8&17\end{bmatrix}$, $\begin{bmatrix}9&6\\24&35\end{bmatrix}$, $\begin{bmatrix}29&8\\4&29\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $C_2^3\times \GU(2,3)$ |
| Contains $-I$: | no $\quad$ (see 40.480.35.bn.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $48$ |
| Full 40-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{139}\cdot5^{64}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2^{6}\cdot4^{2}$ |
| Newforms: | 32.2.a.a, 50.2.a.b$^{5}$, 64.2.b.a, 200.2.a.e$^{3}$, 200.2.d.a$^{2}$, 200.2.d.c$^{2}$, 400.2.a.a$^{2}$, 400.2.a.f$^{2}$, 800.2.a.c, 800.2.a.g, 800.2.a.k, 1600.2.d.c, 1600.2.d.f |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $192$ | $96$ | $0$ | $0$ | full Jacobian |
| 8.192.3-8.j.1.1 | $8$ | $5$ | $5$ | $3$ | $0$ | $1^{14}\cdot2^{5}\cdot4^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.192.3-8.j.1.1 | $8$ | $5$ | $5$ | $3$ | $0$ | $1^{14}\cdot2^{5}\cdot4^{2}$ |
| 40.480.16-40.f.1.1 | $40$ | $2$ | $2$ | $16$ | $1$ | $1^{7}\cdot2^{2}\cdot4^{2}$ |
| 40.480.16-40.f.1.6 | $40$ | $2$ | $2$ | $16$ | $1$ | $1^{7}\cdot2^{2}\cdot4^{2}$ |
| 40.480.17-40.w.2.2 | $40$ | $2$ | $2$ | $17$ | $3$ | $2^{5}\cdot4^{2}$ |
| 40.480.17-40.w.2.10 | $40$ | $2$ | $2$ | $17$ | $3$ | $2^{5}\cdot4^{2}$ |
| 40.480.18-40.a.1.2 | $40$ | $2$ | $2$ | $18$ | $1$ | $1^{7}\cdot2^{5}$ |
| 40.480.18-40.a.1.5 | $40$ | $2$ | $2$ | $18$ | $1$ | $1^{7}\cdot2^{5}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1920.69-40.by.1.1 | $40$ | $2$ | $2$ | $69$ | $13$ | $1^{14}\cdot2^{10}$ |
| 40.1920.69-40.by.3.2 | $40$ | $2$ | $2$ | $69$ | $13$ | $1^{14}\cdot2^{10}$ |
| 40.1920.69-40.bz.1.1 | $40$ | $2$ | $2$ | $69$ | $10$ | $1^{14}\cdot2^{10}$ |
| 40.1920.69-40.bz.2.3 | $40$ | $2$ | $2$ | $69$ | $10$ | $1^{14}\cdot2^{10}$ |
| 40.2880.103-40.fp.1.2 | $40$ | $3$ | $3$ | $103$ | $8$ | $1^{28}\cdot2^{4}\cdot4^{8}$ |
| 40.3840.137-40.gi.1.4 | $40$ | $4$ | $4$ | $137$ | $16$ | $1^{40}\cdot2^{9}\cdot4^{9}\cdot8$ |