Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $576$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $15 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{2}\cdot20^{8}\cdot40^{2}$ | Cusp orbits | $2^{6}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40V15 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.576.15.803 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&38\\16&31\end{bmatrix}$, $\begin{bmatrix}13&20\\4&9\end{bmatrix}$, $\begin{bmatrix}23&24\\16&21\end{bmatrix}$, $\begin{bmatrix}37&10\\16&31\end{bmatrix}$, $\begin{bmatrix}37&22\\24&15\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_4\times C_{10}:C_4^2$ |
| Contains $-I$: | no $\quad$ (see 40.288.15.bf.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $2$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $1280$ |
Jacobian
| Conductor: | $2^{63}\cdot5^{23}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot4^{2}$ |
| Newforms: | 20.2.a.a$^{2}$, 40.2.a.a, 160.2.d.a, 200.2.d.f, 1600.2.a.c, 1600.2.a.k, 1600.2.a.o, 1600.2.a.w |
Rational points
This modular curve has no $\Q_p$ points for $p=7,23,167$, 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 |
|---|---|---|---|---|---|---|
| 40.96.0-40.n.1.3 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.288.7-40.h.1.1 | $40$ | $2$ | $2$ | $7$ | $2$ | $4^{2}$ |
| 40.288.7-40.h.1.37 | $40$ | $2$ | $2$ | $7$ | $2$ | $4^{2}$ |
| 40.288.7-40.q.1.6 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot4$ |
| 40.288.7-40.q.1.57 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot4$ |
| 40.288.7-40.u.2.16 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot4$ |
| 40.288.7-40.u.2.50 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.1152.29-40.dn.3.20 | $40$ | $2$ | $2$ | $29$ | $2$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.dn.4.23 | $40$ | $2$ | $2$ | $29$ | $2$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.dr.2.6 | $40$ | $2$ | $2$ | $29$ | $2$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.dr.3.7 | $40$ | $2$ | $2$ | $29$ | $2$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.en.1.5 | $40$ | $2$ | $2$ | $29$ | $2$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.en.4.5 | $40$ | $2$ | $2$ | $29$ | $2$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.er.1.8 | $40$ | $2$ | $2$ | $29$ | $2$ | $2^{3}\cdot4^{2}$ |
| 40.1152.29-40.er.3.8 | $40$ | $2$ | $2$ | $29$ | $2$ | $2^{3}\cdot4^{2}$ |
| 40.1152.33-40.hd.1.9 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.jr.2.11 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.lw.2.10 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.me.1.11 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.sb.1.5 | $40$ | $2$ | $2$ | $33$ | $2$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.sb.3.9 | $40$ | $2$ | $2$ | $33$ | $2$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.sf.1.10 | $40$ | $2$ | $2$ | $33$ | $2$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.sf.2.11 | $40$ | $2$ | $2$ | $33$ | $2$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.sn.1.10 | $40$ | $2$ | $2$ | $33$ | $2$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.sn.2.11 | $40$ | $2$ | $2$ | $33$ | $2$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.sr.1.5 | $40$ | $2$ | $2$ | $33$ | $2$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.sr.3.9 | $40$ | $2$ | $2$ | $33$ | $2$ | $2^{3}\cdot4^{3}$ |
| 40.1152.33-40.vv.1.11 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.wd.1.5 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.wm.1.5 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.1152.33-40.wq.1.15 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.2880.91-40.fo.1.7 | $40$ | $5$ | $5$ | $91$ | $15$ | $1^{36}\cdot2^{8}\cdot4^{6}$ |