Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}\cdot10^{4}\cdot40^{2}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40M7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.288.7.629 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}15&16\\26&31\end{bmatrix}$, $\begin{bmatrix}17&4\\5&39\end{bmatrix}$, $\begin{bmatrix}19&24\\25&21\end{bmatrix}$, $\begin{bmatrix}23&28\\6&15\end{bmatrix}$, $\begin{bmatrix}23&32\\17&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.144.7.cr.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $2$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $2560$ |
Jacobian
| Conductor: | $2^{20}\cdot5^{11}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 20.2.a.a, 80.2.a.a, 80.2.a.b, 100.2.a.a$^{2}$, 200.2.a.c$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w + x t + y w - y t - w u - t^{2} + t v $ |
| $=$ | $2 x w + y w - z t$ | |
| $=$ | $2 x t - y t - z w$ | |
| $=$ | $2 x^{2} + x y + x z - 2 x u - y^{2} - y z + y u - z t + z v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 125 x^{4} y^{4} z^{2} + 50 x^{4} y^{2} z^{4} - 5 x^{4} z^{6} + 625 x^{2} y^{8} + 500 x^{2} y^{6} z^{2} + \cdots + 400 y^{6} z^{4} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=3,7,43$, and therefore no rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 40.72.3.ff.1 :
| $\displaystyle X$ | $=$ | $\displaystyle 4y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x-2y-w-u+v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2x-2y+w-2t-u+v$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{4}+2X^{3}Y+X^{2}Y^{2}-2XY^{3}-Y^{4}-7X^{3}Z-6X^{2}Z^{2}+2XZ^{3}+Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.7.cr.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{4}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w$ |
Equation of the image curve:
| $0$ | $=$ | $ -125X^{4}Y^{4}Z^{2}+50X^{4}Y^{2}Z^{4}-5X^{4}Z^{6}+625X^{2}Y^{8}+500X^{2}Y^{6}Z^{2}-250X^{2}Y^{4}Z^{4}+20X^{2}Y^{2}Z^{6}+X^{2}Z^{8}+400Y^{6}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.144.3-20.l.1.5 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.48.0-40.w.1.7 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.144.3-20.l.1.22 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.576.13-40.jc.1.13 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.jc.2.13 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.jd.1.5 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.jd.2.5 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.jk.1.5 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.jk.2.5 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.jl.1.7 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.jl.2.7 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.17-40.qi.1.8 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{8}\cdot2$ |
| 40.576.17-40.qj.1.7 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{8}\cdot2$ |
| 40.576.17-40.qk.1.3 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{8}\cdot2$ |
| 40.576.17-40.ql.1.4 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{8}\cdot2$ |
| 40.576.17-40.qm.1.3 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 40.576.17-40.qm.2.4 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 40.576.17-40.qn.1.1 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 40.576.17-40.qn.2.2 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 40.1440.43-40.qo.1.7 | $40$ | $5$ | $5$ | $43$ | $5$ | $1^{36}$ |