Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $480$ | $\PSL_2$-index: | $240$ | ||||
| Genus: | $17 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $20^{4}\cdot40^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 8$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40G17 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.480.17.107 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&0\\34&21\end{bmatrix}$, $\begin{bmatrix}33&28\\26&17\end{bmatrix}$, $\begin{bmatrix}39&4\\24&21\end{bmatrix}$, $\begin{bmatrix}39&36\\22&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.240.17.bm.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{68}\cdot5^{32}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot2^{5}$ |
| Newforms: | 50.2.a.b$^{3}$, 64.2.a.a, 200.2.a.e, 200.2.d.a, 200.2.d.c, 800.2.d.a, 800.2.d.c, 1600.2.a.bb, 1600.2.a.h, 1600.2.a.r |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.m.2.2 | $8$ | $5$ | $5$ | $1$ | $0$ | $1^{6}\cdot2^{5}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.m.2.2 | $8$ | $5$ | $5$ | $1$ | $0$ | $1^{6}\cdot2^{5}$ |
| 40.240.8-40.k.2.8 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{3}\cdot2^{3}$ |
| 40.240.8-40.k.2.11 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{3}\cdot2^{3}$ |
| 40.240.8-40.n.2.4 | $40$ | $2$ | $2$ | $8$ | $0$ | $1^{3}\cdot2^{3}$ |
| 40.240.8-40.n.2.9 | $40$ | $2$ | $2$ | $8$ | $0$ | $1^{3}\cdot2^{3}$ |
| 40.240.9-40.d.1.2 | $40$ | $2$ | $2$ | $9$ | $2$ | $2^{4}$ |
| 40.240.9-40.d.1.20 | $40$ | $2$ | $2$ | $9$ | $2$ | $2^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.960.33-40.bs.2.4 | $40$ | $2$ | $2$ | $33$ | $11$ | $1^{8}\cdot2^{4}$ |
| 40.960.33-40.ct.2.4 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{8}\cdot2^{4}$ |
| 40.960.33-40.dc.2.4 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{8}\cdot2^{4}$ |
| 40.960.33-40.de.2.4 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{8}\cdot2^{4}$ |
| 40.960.33-40.dk.2.4 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{8}\cdot2^{4}$ |
| 40.960.33-40.dm.2.4 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{8}\cdot2^{4}$ |
| 40.960.33-40.dq.2.4 | $40$ | $2$ | $2$ | $33$ | $11$ | $1^{8}\cdot2^{4}$ |
| 40.960.33-40.dt.2.4 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{8}\cdot2^{4}$ |
| 40.1440.49-40.ee.2.5 | $40$ | $3$ | $3$ | $49$ | $6$ | $1^{12}\cdot2^{2}\cdot4^{4}$ |
| 40.1920.65-40.hx.2.5 | $40$ | $4$ | $4$ | $65$ | $13$ | $1^{16}\cdot2^{8}\cdot4^{4}$ |