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: | 40.480.15.9 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&4\\0&1\end{bmatrix}$, $\begin{bmatrix}15&12\\18&23\end{bmatrix}$, $\begin{bmatrix}23&36\\8&17\end{bmatrix}$, $\begin{bmatrix}25&8\\22&7\end{bmatrix}$, $\begin{bmatrix}31&28\\12&33\end{bmatrix}$, $\begin{bmatrix}37&4\\6&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.240.15.z.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: | $2^{37}\cdot5^{30}$ |
| 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.2.13 | $8$ | $10$ | $10$ | $0$ | $0$ | 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.2.13 | $8$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 20.240.7-20.b.1.12 | $20$ | $2$ | $2$ | $7$ | $0$ | $2^{2}\cdot4$ |
| 40.240.7-20.b.1.19 | $40$ | $2$ | $2$ | $7$ | $0$ | $2^{2}\cdot4$ |
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.1.26 | $40$ | $2$ | $2$ | $29$ | $1$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.ed.1.26 | $40$ | $2$ | $2$ | $29$ | $6$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.eh.1.26 | $40$ | $2$ | $2$ | $29$ | $3$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.el.1.26 | $40$ | $2$ | $2$ | $29$ | $0$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.ep.1.28 | $40$ | $2$ | $2$ | $29$ | $1$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.et.1.26 | $40$ | $2$ | $2$ | $29$ | $4$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.ex.1.29 | $40$ | $2$ | $2$ | $29$ | $3$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.29-40.fb.1.26 | $40$ | $2$ | $2$ | $29$ | $4$ | $1^{6}\cdot2^{2}\cdot4$ |
| 40.960.31-40.h.1.1 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.l.1.13 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.t.1.7 | $40$ | $2$ | $2$ | $31$ | $6$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.x.1.7 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.bh.2.15 | $40$ | $2$ | $2$ | $31$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.bj.1.3 | $40$ | $2$ | $2$ | $31$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.bp.1.7 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.br.1.7 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.bx.1.32 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.bz.2.30 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.cf.2.31 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.ch.1.32 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.cn.2.26 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.cp.1.31 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.cv.1.32 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.cx.2.31 | $40$ | $2$ | $2$ | $31$ | $6$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.dd.1.13 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.df.2.13 | $40$ | $2$ | $2$ | $31$ | $6$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.dl.2.9 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.dn.1.15 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.dt.2.13 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.dv.1.13 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.eb.1.15 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.ed.2.9 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.ej.1.20 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.el.1.24 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.er.1.18 | $40$ | $2$ | $2$ | $31$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.et.1.20 | $40$ | $2$ | $2$ | $31$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.fb.1.24 | $40$ | $2$ | $2$ | $31$ | $6$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.ff.1.20 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.fn.1.24 | $40$ | $2$ | $2$ | $31$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.31-40.fr.1.20 | $40$ | $2$ | $2$ | $31$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 40.960.33-40.fh.1.20 | $40$ | $2$ | $2$ | $33$ | $5$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.fp.1.18 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.gg.1.24 | $40$ | $2$ | $2$ | $33$ | $6$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.gk.1.20 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.ix.1.20 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.iz.1.24 | $40$ | $2$ | $2$ | $33$ | $3$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.jf.1.20 | $40$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.jh.1.24 | $40$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{4}\cdot4$ |
| 40.960.33-40.md.1.30 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.mf.2.29 | $40$ | $2$ | $2$ | $33$ | $6$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.ml.2.26 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.mn.1.31 | $40$ | $2$ | $2$ | $33$ | $6$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.mt.2.29 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.mv.1.30 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.nb.1.31 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.960.33-40.nd.1.26 | $40$ | $2$ | $2$ | $33$ | $4$ | $1^{4}\cdot2^{5}\cdot4$ |
| 40.1440.43-40.fl.1.48 | $40$ | $3$ | $3$ | $43$ | $1$ | $1^{12}\cdot2^{2}\cdot4^{3}$ |