Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $1440$ | $\PSL_2$-index: | $1440$ | ||||
| Genus: | $101 = 1 + \frac{ 1440 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 36 }{2}$ | ||||||
| Cusps: | $36$ (none of which are rational) | Cusp widths | $40^{36}$ | Cusp orbits | $4^{5}\cdot8^{2}$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $42$ | ||||||
| $\Q$-gonality: | $26 \le \gamma \le 32$ | ||||||
| $\overline{\Q}$-gonality: | $26 \le \gamma \le 32$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.1440.101.75 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&23\\20&37\end{bmatrix}$, $\begin{bmatrix}7&19\\6&33\end{bmatrix}$, $\begin{bmatrix}11&35\\10&21\end{bmatrix}$, $\begin{bmatrix}19&28\\0&3\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $8$ |
| Cyclic 40-torsion field degree: | $128$ |
| Full 40-torsion field degree: | $512$ |
Jacobian
| Conductor: | $2^{504}\cdot5^{178}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{83}\cdot2^{9}$ |
| Newforms: | 20.2.a.a$^{2}$, 32.2.a.a, 40.2.a.a$^{2}$, 50.2.a.a, 50.2.a.b$^{2}$, 80.2.a.a$^{3}$, 80.2.a.b$^{2}$, 100.2.a.a$^{2}$, 160.2.a.a$^{2}$, 160.2.a.b$^{2}$, 160.2.a.c, 200.2.a.a, 200.2.a.c$^{2}$, 200.2.a.e, 320.2.a.a, 320.2.a.b, 320.2.a.c$^{2}$, 320.2.a.d, 320.2.a.e, 320.2.a.f, 400.2.a.a, 400.2.a.b, 400.2.a.c, 400.2.a.d$^{2}$, 400.2.a.e$^{2}$, 400.2.a.g, 400.2.a.h$^{2}$, 800.2.a.a, 800.2.a.b$^{2}$, 800.2.a.d$^{2}$, 800.2.a.h$^{2}$, 800.2.a.i, 800.2.a.j, 800.2.a.l, 800.2.a.m, 800.2.a.n, 1600.2.a.a$^{2}$, 1600.2.a.b, 1600.2.a.bb, 1600.2.a.bc, 1600.2.a.bd, 1600.2.a.c$^{2}$, 1600.2.a.d$^{2}$, 1600.2.a.e$^{2}$, 1600.2.a.f$^{2}$, 1600.2.a.h$^{2}$, 1600.2.a.i, 1600.2.a.k, 1600.2.a.l, 1600.2.a.m, 1600.2.a.n, 1600.2.a.o$^{2}$, 1600.2.a.p$^{2}$, 1600.2.a.q$^{2}$, 1600.2.a.r$^{2}$, 1600.2.a.t$^{2}$, 1600.2.a.u$^{2}$, 1600.2.a.v$^{2}$, 1600.2.a.w, 1600.2.a.x, 1600.2.a.y, 1600.2.a.z |
Rational points
This modular curve has real points and $\Q_p$ points for good $p < 8192$, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.720.47.sv.1 | $40$ | $2$ | $2$ | $47$ | $23$ | $1^{40}\cdot2^{7}$ |
| 40.720.49.evr.1 | $40$ | $2$ | $2$ | $49$ | $18$ | $1^{40}\cdot2^{6}$ |
| 40.720.49.evt.1 | $40$ | $2$ | $2$ | $49$ | $23$ | $1^{42}\cdot2^{5}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.2880.205.in.2 | $40$ | $2$ | $2$ | $205$ | $71$ | $1^{80}\cdot2^{12}$ |
| 40.2880.205.bnl.1 | $40$ | $2$ | $2$ | $205$ | $77$ | $1^{80}\cdot2^{12}$ |
| 40.2880.205.boz.1 | $40$ | $2$ | $2$ | $205$ | $73$ | $1^{80}\cdot2^{12}$ |
| 40.2880.205.bpj.1 | $40$ | $2$ | $2$ | $205$ | $79$ | $1^{80}\cdot2^{12}$ |
| 40.2880.205.cbj.1 | $40$ | $2$ | $2$ | $205$ | $71$ | $1^{80}\cdot2^{12}$ |
| 40.2880.205.cbs.1 | $40$ | $2$ | $2$ | $205$ | $72$ | $1^{80}\cdot2^{12}$ |
| 40.2880.205.ccz.1 | $40$ | $2$ | $2$ | $205$ | $76$ | $1^{80}\cdot2^{12}$ |
| 40.2880.205.cdk.1 | $40$ | $2$ | $2$ | $205$ | $77$ | $1^{80}\cdot2^{12}$ |