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 | $4^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $4$ | ||||||
$\Q$-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: | 40.480.15.2668 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}13&16\\14&7\end{bmatrix}$, $\begin{bmatrix}21&36\\5&39\end{bmatrix}$, $\begin{bmatrix}25&28\\27&27\end{bmatrix}$, $\begin{bmatrix}27&4\\37&23\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.240.15.gm.1 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: | $2^{52}\cdot5^{26}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{15}$ |
Newforms: | 50.2.a.b$^{2}$, 80.2.a.a$^{2}$, 80.2.a.b$^{2}$, 100.2.a.a, 400.2.a.a, 400.2.a.c, 400.2.a.d$^{2}$, 400.2.a.e, 400.2.a.f, 400.2.a.h$^{2}$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=3$, 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 |
---|---|---|---|---|---|---|
40.48.0-40.bw.1.1 | $40$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
40.240.7-20.t.1.2 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{8}$ |
40.240.7-20.t.1.7 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{8}$ |
40.240.7-40.cu.1.4 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{8}$ |
40.240.7-40.cu.1.30 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{8}$ |
40.240.7-40.cw.1.3 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{8}$ |
40.240.7-40.cw.1.29 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{8}$ |
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.vw.1.3 | $40$ | $2$ | $2$ | $29$ | $7$ | $1^{14}$ |
40.960.29-40.vy.1.2 | $40$ | $2$ | $2$ | $29$ | $10$ | $1^{14}$ |
40.960.29-40.we.1.5 | $40$ | $2$ | $2$ | $29$ | $9$ | $1^{14}$ |
40.960.29-40.wg.1.2 | $40$ | $2$ | $2$ | $29$ | $12$ | $1^{14}$ |
40.960.29-40.yu.1.2 | $40$ | $2$ | $2$ | $29$ | $6$ | $1^{14}$ |
40.960.29-40.yw.1.3 | $40$ | $2$ | $2$ | $29$ | $7$ | $1^{14}$ |
40.960.29-40.zc.1.1 | $40$ | $2$ | $2$ | $29$ | $16$ | $1^{14}$ |
40.960.29-40.ze.1.2 | $40$ | $2$ | $2$ | $29$ | $7$ | $1^{14}$ |
40.1440.43-40.bfy.1.3 | $40$ | $3$ | $3$ | $43$ | $10$ | $1^{28}$ |