Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $800$ | ||
| Index: | $1152$ | $\PSL_2$-index: | $576$ | ||||
| Genus: | $33 = 1 + \frac{ 576 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}\cdot20^{8}\cdot40^{8}$ | Cusp orbits | $2^{4}\cdot4^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $7 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $7 \le \gamma \le 12$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.1152.33.3616 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&12\\30&27\end{bmatrix}$, $\begin{bmatrix}9&16\\20&1\end{bmatrix}$, $\begin{bmatrix}11&32\\30&1\end{bmatrix}$, $\begin{bmatrix}17&8\\0&33\end{bmatrix}$, $\begin{bmatrix}31&32\\20&19\end{bmatrix}$ |
| $\GL_2(\Z/40\Z)$-subgroup: | $D_{10}.C_2^5$ |
| Contains $-I$: | no $\quad$ (see 40.576.33.gy.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $2$ |
| Cyclic 40-torsion field degree: | $16$ |
| Full 40-torsion field degree: | $640$ |
Jacobian
| Conductor: | $2^{135}\cdot5^{51}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2\cdot4^{4}$ |
| Newforms: | 20.2.a.a$^{3}$, 40.2.a.a$^{2}$, 80.2.a.a, 80.2.a.b, 100.2.a.a, 160.2.d.a$^{2}$, 200.2.a.c, 200.2.d.f, 400.2.a.c, 400.2.a.e, 800.2.a.a, 800.2.a.d$^{2}$, 800.2.a.i, 800.2.a.m, 800.2.d.e |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=7,23,79,167$, 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 |
|---|---|---|---|---|---|---|
| 40.192.1-40.b.2.2 | $40$ | $6$ | $6$ | $1$ | $1$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.576.15-40.a.1.13 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.576.15-40.a.1.21 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.576.15-40.e.1.5 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.576.15-40.e.1.18 | $40$ | $2$ | $2$ | $15$ | $0$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.576.15-40.db.1.12 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.576.15-40.db.1.23 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.576.15-40.dd.1.14 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.576.15-40.dd.1.27 | $40$ | $2$ | $2$ | $15$ | $1$ | $1^{8}\cdot2\cdot4^{2}$ |
| 40.576.17-40.bv.1.3 | $40$ | $2$ | $2$ | $17$ | $4$ | $4^{4}$ |
| 40.576.17-40.bv.1.13 | $40$ | $2$ | $2$ | $17$ | $4$ | $4^{4}$ |
| 40.576.17-40.em.2.2 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{8}\cdot4^{2}$ |
| 40.576.17-40.em.2.28 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{8}\cdot4^{2}$ |
| 40.576.17-40.eo.2.7 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{8}\cdot4^{2}$ |
| 40.576.17-40.eo.2.30 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{8}\cdot4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.2304.65-40.gq.3.3 | $40$ | $2$ | $2$ | $65$ | $4$ | $2^{6}\cdot4^{5}$ |
| 40.2304.65-40.gq.4.2 | $40$ | $2$ | $2$ | $65$ | $4$ | $2^{6}\cdot4^{5}$ |
| 40.2304.65-40.gu.3.5 | $40$ | $2$ | $2$ | $65$ | $4$ | $2^{6}\cdot4^{5}$ |
| 40.2304.65-40.gu.4.3 | $40$ | $2$ | $2$ | $65$ | $4$ | $2^{6}\cdot4^{5}$ |
| 40.2304.65-40.uc.1.1 | $40$ | $2$ | $2$ | $65$ | $4$ | $2^{6}\cdot4^{5}$ |
| 40.2304.65-40.uc.2.1 | $40$ | $2$ | $2$ | $65$ | $4$ | $2^{6}\cdot4^{5}$ |
| 40.2304.65-40.ug.1.5 | $40$ | $2$ | $2$ | $65$ | $4$ | $2^{6}\cdot4^{5}$ |
| 40.2304.65-40.ug.2.3 | $40$ | $2$ | $2$ | $65$ | $4$ | $2^{6}\cdot4^{5}$ |
| 40.2304.73-40.la.1.4 | $40$ | $2$ | $2$ | $73$ | $10$ | $1^{16}\cdot2^{4}\cdot4^{4}$ |
| 40.2304.73-40.lc.2.4 | $40$ | $2$ | $2$ | $73$ | $9$ | $1^{16}\cdot2^{4}\cdot4^{4}$ |
| 40.2304.73-40.lh.2.4 | $40$ | $2$ | $2$ | $73$ | $10$ | $1^{16}\cdot2^{4}\cdot4^{4}$ |
| 40.2304.73-40.lj.4.4 | $40$ | $2$ | $2$ | $73$ | $9$ | $1^{16}\cdot2^{4}\cdot4^{4}$ |
| 40.2304.73-40.lo.1.4 | $40$ | $2$ | $2$ | $73$ | $4$ | $2^{6}\cdot4^{5}\cdot8$ |
| 40.2304.73-40.lo.2.4 | $40$ | $2$ | $2$ | $73$ | $4$ | $2^{6}\cdot4^{5}\cdot8$ |
| 40.2304.73-40.lo.3.4 | $40$ | $2$ | $2$ | $73$ | $4$ | $2^{6}\cdot4^{5}\cdot8$ |
| 40.2304.73-40.lo.4.4 | $40$ | $2$ | $2$ | $73$ | $4$ | $2^{6}\cdot4^{5}\cdot8$ |
| 40.5760.193-40.lf.1.2 | $40$ | $5$ | $5$ | $193$ | $27$ | $1^{68}\cdot2^{22}\cdot4^{12}$ |