Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| 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 | $2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-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: | 40.480.15.2468 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&24\\13&31\end{bmatrix}$, $\begin{bmatrix}13&16\\6&1\end{bmatrix}$, $\begin{bmatrix}13&16\\33&27\end{bmatrix}$, $\begin{bmatrix}17&8\\23&23\end{bmatrix}$, $\begin{bmatrix}25&24\\16&25\end{bmatrix}$, $\begin{bmatrix}39&24\\23&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.240.15.dh.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{38}\cdot5^{30}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}$ |
| Newforms: | 50.2.a.b$^{4}$, 100.2.a.a$^{3}$, 200.2.a.c$^{2}$, 200.2.a.e$^{2}$, 400.2.a.a, 400.2.a.c, 400.2.a.e, 400.2.a.f |
Rational points
This modular curve has no $\Q_p$ points for $p=3,17$, 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 |
|---|---|---|---|---|---|---|
| 40.48.0-8.q.1.4 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 40.240.7-20.p.1.6 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{8}$ |
| 40.240.7-20.p.1.15 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{8}$ |
| 40.240.7-40.cj.1.10 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{8}$ |
| 40.240.7-40.cj.1.12 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{8}$ |
| 40.240.7-40.cj.1.45 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{8}$ |
| 40.240.7-40.cj.1.47 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{8}$ |
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.li.1.4 | $40$ | $2$ | $2$ | $29$ | $5$ | $1^{14}$ |
| 40.960.29-40.lj.1.1 | $40$ | $2$ | $2$ | $29$ | $12$ | $1^{14}$ |
| 40.960.29-40.lq.1.7 | $40$ | $2$ | $2$ | $29$ | $5$ | $1^{14}$ |
| 40.960.29-40.lr.1.5 | $40$ | $2$ | $2$ | $29$ | $8$ | $1^{14}$ |
| 40.960.29-40.ly.1.5 | $40$ | $2$ | $2$ | $29$ | $10$ | $1^{14}$ |
| 40.960.29-40.lz.1.10 | $40$ | $2$ | $2$ | $29$ | $5$ | $1^{14}$ |
| 40.960.29-40.mg.1.1 | $40$ | $2$ | $2$ | $29$ | $6$ | $1^{14}$ |
| 40.960.29-40.mh.1.1 | $40$ | $2$ | $2$ | $29$ | $7$ | $1^{14}$ |
| 40.960.31-40.fw.1.4 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fw.1.6 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fw.2.4 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fw.2.8 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fx.1.3 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fx.1.5 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fx.2.1 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fx.2.5 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fy.1.5 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fy.1.9 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fy.2.1 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fy.2.9 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fz.1.7 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fz.1.11 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fz.2.7 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fz.2.15 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.33-40.xg.1.5 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{14}\cdot2^{2}$ |
| 40.960.33-40.xh.1.5 | $40$ | $2$ | $2$ | $33$ | $10$ | $1^{14}\cdot2^{2}$ |
| 40.960.33-40.xi.1.7 | $40$ | $2$ | $2$ | $33$ | $6$ | $1^{14}\cdot2^{2}$ |
| 40.960.33-40.xj.1.5 | $40$ | $2$ | $2$ | $33$ | $8$ | $1^{14}\cdot2^{2}$ |
| 40.960.33-40.xk.1.3 | $40$ | $2$ | $2$ | $33$ | $10$ | $1^{12}\cdot2^{3}$ |
| 40.960.33-40.xl.1.4 | $40$ | $2$ | $2$ | $33$ | $8$ | $1^{12}\cdot2^{3}$ |
| 40.960.33-40.xm.1.3 | $40$ | $2$ | $2$ | $33$ | $10$ | $1^{12}\cdot2^{3}$ |
| 40.960.33-40.xn.1.3 | $40$ | $2$ | $2$ | $33$ | $8$ | $1^{12}\cdot2^{3}$ |
| 40.1440.43-40.om.1.15 | $40$ | $3$ | $3$ | $43$ | $6$ | $1^{28}$ |