Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $480$ | $\PSL_2$-index: | $240$ | ||||
| Genus: | $15 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $10^{8}\cdot40^{4}$ | Cusp orbits | $4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $6$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40C15 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.480.15.2664 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&36\\13&17\end{bmatrix}$, $\begin{bmatrix}11&16\\2&29\end{bmatrix}$, $\begin{bmatrix}15&16\\17&15\end{bmatrix}$, $\begin{bmatrix}27&36\\7&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.240.15.gn.1 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: | $2^{68}\cdot5^{26}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}$ |
| Newforms: | 50.2.a.b$^{2}$, 80.2.a.a, 80.2.a.b, 100.2.a.a, 320.2.a.c, 320.2.a.f, 400.2.a.d, 400.2.a.h, 1600.2.a.a, 1600.2.a.c, 1600.2.a.j, 1600.2.a.o, 1600.2.a.q, 1600.2.a.x |
Rational points
This modular curve has no $\Q_p$ points for $p=3$, 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 |
|---|---|---|---|---|---|---|
| 40.48.0-40.bx.1.2 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 40.240.7-20.t.1.2 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{8}$ |
| 40.240.7-20.t.1.11 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{8}$ |
| 40.240.7-40.cv.1.13 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{8}$ |
| 40.240.7-40.cv.1.19 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{8}$ |
| 40.240.7-40.cx.1.14 | $40$ | $2$ | $2$ | $7$ | $4$ | $1^{8}$ |
| 40.240.7-40.cx.1.20 | $40$ | $2$ | $2$ | $7$ | $4$ | $1^{8}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.960.29-40.vx.1.4 | $40$ | $2$ | $2$ | $29$ | $9$ | $1^{14}$ |
| 40.960.29-40.vz.1.4 | $40$ | $2$ | $2$ | $29$ | $12$ | $1^{14}$ |
| 40.960.29-40.wf.1.6 | $40$ | $2$ | $2$ | $29$ | $15$ | $1^{14}$ |
| 40.960.29-40.wh.1.4 | $40$ | $2$ | $2$ | $29$ | $10$ | $1^{14}$ |
| 40.960.29-40.yv.1.6 | $40$ | $2$ | $2$ | $29$ | $8$ | $1^{14}$ |
| 40.960.29-40.yx.1.6 | $40$ | $2$ | $2$ | $29$ | $9$ | $1^{14}$ |
| 40.960.29-40.zd.1.6 | $40$ | $2$ | $2$ | $29$ | $14$ | $1^{14}$ |
| 40.960.29-40.zf.1.7 | $40$ | $2$ | $2$ | $29$ | $13$ | $1^{14}$ |
| 40.1440.43-40.bfz.1.6 | $40$ | $3$ | $3$ | $43$ | $16$ | $1^{28}$ |