Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $200$ | ||
| 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 | $10^{4}\cdot20^{6}\cdot40^{2}$ | Cusp orbits | $2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40O15 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.480.15.103 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&4\\18&29\end{bmatrix}$, $\begin{bmatrix}7&8\\32&9\end{bmatrix}$, $\begin{bmatrix}7&20\\0&7\end{bmatrix}$, $\begin{bmatrix}9&20\\16&31\end{bmatrix}$, $\begin{bmatrix}23&8\\10&27\end{bmatrix}$, $\begin{bmatrix}39&32\\8&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.240.15.z.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{37}\cdot5^{30}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot2^{2}\cdot4$ |
| Newforms: | 50.2.a.b$^{3}$, 100.2.a.a$^{2}$, 200.2.a.c, 200.2.a.e, 200.2.d.a, 200.2.d.c, 200.2.d.f |
Rational points
This modular curve has no $\Q_p$ points for $p=3$, and therefore no 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.e.1.15 | $8$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.e.1.15 | $8$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 40.240.7-20.b.1.8 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}\cdot4$ |
| 40.240.7-20.b.1.32 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.960.29-40.dz.2.11 | $40$ | $2$ | $2$ | $29$ | $1$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.ed.2.11 | $40$ | $2$ | $2$ | $29$ | $6$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.eh.2.13 | $40$ | $2$ | $2$ | $29$ | $3$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.el.2.14 | $40$ | $2$ | $2$ | $29$ | $0$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.ep.2.15 | $40$ | $2$ | $2$ | $29$ | $1$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.et.2.13 | $40$ | $2$ | $2$ | $29$ | $4$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.ex.2.13 | $40$ | $2$ | $2$ | $29$ | $3$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.fb.2.9 | $40$ | $2$ | $2$ | $29$ | $4$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.31-40.h.2.27 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.l.1.25 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.t.2.31 | $40$ | $2$ | $2$ | $31$ | $6$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.x.1.28 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.bh.1.31 | $40$ | $2$ | $2$ | $31$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.bj.2.27 | $40$ | $2$ | $2$ | $31$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.bp.2.26 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.br.2.31 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.bx.2.23 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.bz.1.19 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.cf.1.22 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.ch.2.21 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.cn.1.21 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.cp.2.23 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.cv.2.17 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.cx.1.21 | $40$ | $2$ | $2$ | $31$ | $6$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.dd.2.27 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.df.1.29 | $40$ | $2$ | $2$ | $31$ | $6$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.dl.1.29 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.dn.2.26 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.dt.1.30 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.dv.2.31 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.eb.2.26 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.ed.1.29 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.ej.2.26 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.el.2.20 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.er.2.30 | $40$ | $2$ | $2$ | $31$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.et.2.27 | $40$ | $2$ | $2$ | $31$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.fb.2.24 | $40$ | $2$ | $2$ | $31$ | $6$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.ff.1.28 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.fn.2.23 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.fr.2.24 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.33-40.fh.2.12 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.fp.2.2 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.gg.2.9 | $40$ | $2$ | $2$ | $33$ | $6$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.gk.1.7 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.ix.2.5 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.iz.2.10 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.jf.2.5 | $40$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.jh.2.13 | $40$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.md.2.10 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.mf.1.11 | $40$ | $2$ | $2$ | $33$ | $6$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.ml.1.14 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.mn.2.9 | $40$ | $2$ | $2$ | $33$ | $6$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.mt.1.12 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.mv.2.14 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.nb.2.10 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.nd.2.16 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.1440.43-40.fl.2.24 | $40$ | $3$ | $3$ | $43$ | $1$ | $1^{12}\cdot2^{2}\cdot4^{3}$ |