Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $480$ | $\PSL_2$-index: | $480$ | ||||
| Genus: | $33 = 1 + \frac{ 480 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $20^{8}\cdot40^{8}$ | Cusp orbits | $4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $8$ | ||||||
| $\Q$-gonality: | $10 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $10 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.480.33.349 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&16\\8&7\end{bmatrix}$, $\begin{bmatrix}9&12\\28&17\end{bmatrix}$, $\begin{bmatrix}15&13\\32&17\end{bmatrix}$, $\begin{bmatrix}35&13\\8&5\end{bmatrix}$, $\begin{bmatrix}37&15\\0&27\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 40.960.33-40.bka.1.1, 40.960.33-40.bka.1.2, 40.960.33-40.bka.1.3, 40.960.33-40.bka.1.4, 40.960.33-40.bka.1.5, 40.960.33-40.bka.1.6, 40.960.33-40.bka.1.7, 40.960.33-40.bka.1.8 |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{153}\cdot5^{56}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{29}\cdot2^{2}$ |
| Newforms: | 50.2.a.b$^{3}$, 64.2.a.a, 100.2.a.a$^{2}$, 200.2.a.c, 200.2.a.e, 320.2.a.b, 320.2.a.c$^{2}$, 320.2.a.e, 320.2.a.f$^{2}$, 320.2.a.g, 400.2.a.a$^{2}$, 400.2.a.c$^{2}$, 400.2.a.e$^{2}$, 400.2.a.f$^{2}$, 1600.2.a.ba, 1600.2.a.g, 1600.2.a.j$^{2}$, 1600.2.a.n, 1600.2.a.s, 1600.2.a.x$^{2}$ |
Rational points
This modular curve has no $\Q_p$ points for $p=3,13,17,23,157,293$, 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 |
|---|---|---|---|---|---|---|
| 20.240.15.p.1 | $20$ | $2$ | $2$ | $15$ | $4$ | $1^{14}\cdot2^{2}$ |
| 40.48.1.ea.1 | $40$ | $10$ | $10$ | $1$ | $0$ | $1^{28}\cdot2^{2}$ |
| 40.240.15.ez.1 | $40$ | $2$ | $2$ | $15$ | $4$ | $1^{14}\cdot2^{2}$ |
| 40.240.15.za.1 | $40$ | $2$ | $2$ | $15$ | $5$ | $1^{14}\cdot2^{2}$ |
| 40.240.15.zb.1 | $40$ | $2$ | $2$ | $15$ | $3$ | $1^{14}\cdot2^{2}$ |
| 40.240.17.hf.1 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{16}$ |
| 40.240.17.uu.1 | $40$ | $2$ | $2$ | $17$ | $7$ | $1^{16}$ |
| 40.240.17.uv.1 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{16}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.960.65.cls.1 | $40$ | $2$ | $2$ | $65$ | $19$ | $1^{26}\cdot2^{3}$ |
| 40.960.65.clw.1 | $40$ | $2$ | $2$ | $65$ | $20$ | $1^{26}\cdot2^{3}$ |
| 40.960.65.cmy.1 | $40$ | $2$ | $2$ | $65$ | $15$ | $1^{26}\cdot2^{3}$ |
| 40.960.65.cnc.1 | $40$ | $2$ | $2$ | $65$ | $20$ | $1^{26}\cdot2^{3}$ |
| 40.960.65.cqq.1 | $40$ | $2$ | $2$ | $65$ | $22$ | $1^{26}\cdot2^{3}$ |
| 40.960.65.cqu.1 | $40$ | $2$ | $2$ | $65$ | $23$ | $1^{26}\cdot2^{3}$ |
| 40.960.65.crw.1 | $40$ | $2$ | $2$ | $65$ | $22$ | $1^{26}\cdot2^{3}$ |
| 40.960.65.csa.1 | $40$ | $2$ | $2$ | $65$ | $21$ | $1^{26}\cdot2^{3}$ |
| 40.1440.97.dpe.1 | $40$ | $3$ | $3$ | $97$ | $25$ | $1^{54}\cdot2^{5}$ |