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: | 40.480.15.35 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&24\\6&21\end{bmatrix}$, $\begin{bmatrix}17&24\\2&33\end{bmatrix}$, $\begin{bmatrix}21&16\\34&15\end{bmatrix}$, $\begin{bmatrix}25&32\\12&5\end{bmatrix}$, $\begin{bmatrix}29&8\\22&31\end{bmatrix}$, $\begin{bmatrix}29&8\\34&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.240.15.ch.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
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.i.1.2 | $8$ | $10$ | $10$ | $0$ | $0$ | 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.i.1.2 | $8$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 20.240.7-20.b.1.3 | $20$ | $2$ | $2$ | $7$ | $0$ | $1^{8}$ |
| 40.240.7-20.b.1.44 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{8}$ |
| 40.240.7-40.cj.1.27 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{8}$ |
| 40.240.7-40.cj.1.38 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{8}$ |
| 40.240.7-40.cw.1.11 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{8}$ |
| 40.240.7-40.cw.1.22 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{8}$ |
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.go.1.1 | $40$ | $2$ | $2$ | $29$ | $5$ | $1^{14}$ |
| 40.960.29-40.gp.1.9 | $40$ | $2$ | $2$ | $29$ | $12$ | $1^{14}$ |
| 40.960.29-40.gw.1.11 | $40$ | $2$ | $2$ | $29$ | $7$ | $1^{14}$ |
| 40.960.29-40.gx.1.9 | $40$ | $2$ | $2$ | $29$ | $6$ | $1^{14}$ |
| 40.960.29-40.he.1.12 | $40$ | $2$ | $2$ | $29$ | $5$ | $1^{14}$ |
| 40.960.29-40.hf.1.9 | $40$ | $2$ | $2$ | $29$ | $8$ | $1^{14}$ |
| 40.960.29-40.hm.1.1 | $40$ | $2$ | $2$ | $29$ | $5$ | $1^{14}$ |
| 40.960.29-40.hn.1.9 | $40$ | $2$ | $2$ | $29$ | $10$ | $1^{14}$ |
| 40.960.31-40.fg.1.1 | $40$ | $2$ | $2$ | $31$ | $6$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fg.2.1 | $40$ | $2$ | $2$ | $31$ | $6$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fh.1.3 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fh.2.5 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fi.1.1 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fi.2.1 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fj.1.3 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fj.2.5 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fk.1.1 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fk.2.1 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fl.1.2 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fl.2.3 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fm.1.3 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fm.2.5 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fn.1.1 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fn.2.1 | $40$ | $2$ | $2$ | $31$ | $4$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fo.1.2 | $40$ | $2$ | $2$ | $31$ | $6$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fo.2.3 | $40$ | $2$ | $2$ | $31$ | $6$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fp.1.1 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fp.2.1 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fq.1.5 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fq.2.9 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fr.1.1 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.31-40.fr.2.1 | $40$ | $2$ | $2$ | $31$ | $2$ | $2^{4}\cdot4^{2}$ |
| 40.960.33-40.eu.1.1 | $40$ | $2$ | $2$ | $33$ | $7$ | $1^{14}\cdot2^{2}$ |
| 40.960.33-40.gv.1.1 | $40$ | $2$ | $2$ | $33$ | $10$ | $1^{14}\cdot2^{2}$ |
| 40.960.33-40.lw.1.1 | $40$ | $2$ | $2$ | $33$ | $6$ | $1^{14}\cdot2^{2}$ |
| 40.960.33-40.lx.1.1 | $40$ | $2$ | $2$ | $33$ | $8$ | $1^{14}\cdot2^{2}$ |
| 40.960.33-40.om.1.1 | $40$ | $2$ | $2$ | $33$ | $10$ | $1^{12}\cdot2^{3}$ |
| 40.960.33-40.on.1.1 | $40$ | $2$ | $2$ | $33$ | $8$ | $1^{12}\cdot2^{3}$ |
| 40.960.33-40.oy.1.1 | $40$ | $2$ | $2$ | $33$ | $10$ | $1^{12}\cdot2^{3}$ |
| 40.960.33-40.oz.1.1 | $40$ | $2$ | $2$ | $33$ | $8$ | $1^{12}\cdot2^{3}$ |
| 40.1440.43-40.jg.1.12 | $40$ | $3$ | $3$ | $43$ | $6$ | $1^{28}$ |