Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $80$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $8$ are rational) | Cusp widths | $2^{4}\cdot8^{2}\cdot10^{4}\cdot40^{2}$ | Cusp orbits | $1^{8}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40M7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.288.7.3332 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}17&4\\0&17\end{bmatrix}$, $\begin{bmatrix}17&21\\0&39\end{bmatrix}$, $\begin{bmatrix}23&16\\0&3\end{bmatrix}$, $\begin{bmatrix}29&36\\0&29\end{bmatrix}$, $\begin{bmatrix}31&29\\0&37\end{bmatrix}$, $\begin{bmatrix}33&2\\0&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.144.7.cl.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $1$ |
| Cyclic 40-torsion field degree: | $16$ |
| Full 40-torsion field degree: | $2560$ |
Jacobian
| Conductor: | $2^{20}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 20.2.a.a$^{3}$, 40.2.a.a$^{2}$, 80.2.a.a, 80.2.a.b |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ y v + t^{2} $ |
| $=$ | $y v - z u$ | |
| $=$ | $2 x u - w u + t v$ | |
| $=$ | $2 x v + w v + t u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{2} z^{2} - x^{2} y^{6} + x^{2} y^{4} z^{2} + x^{2} y^{2} z^{4} - x^{2} z^{6} + y^{8} + \cdots + z^{8} $ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(-1/4:0:0:-1/2:0:1:0)$, $(1/4:0:0:-1/2:0:0:1)$, $(-1/2:0:-2:1:0:0:0)$, $(1/4:0:0:1/2:0:1:0)$, $(1/2:-2:0:1:0:0:0)$, $(-1/4:0:0:1/2:0:0:1)$, $(1/2:2:0:1:0:0:0)$, $(-1/2:0:2:1:0:0:0)$ |
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.fd.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y+z+2w-u-v$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -4w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y-z-2w-u-v$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+2X^{3}Y+X^{2}Y^{2}-Y^{3}Z+2YZ^{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.cl.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{4}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{2}Z^{2}-X^{2}Y^{6}+X^{2}Y^{4}Z^{2}+X^{2}Y^{2}Z^{4}-X^{2}Z^{6}+Y^{8}+4Y^{6}Z^{2}+6Y^{4}Z^{4}+4Y^{2}Z^{6}+Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.0-8.q.1.4 | $40$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 40.144.3-20.l.1.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.144.3-20.l.1.5 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.144.3-40.bx.1.14 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.144.3-40.bx.1.19 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.144.3-40.bx.1.39 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 40.144.3-40.bx.1.42 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
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.hk.1.10 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.hk.2.14 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.hl.1.2 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.hl.2.4 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.hs.1.6 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.hs.2.2 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.ht.1.6 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.13-40.ht.2.8 | $40$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 40.576.15-40.ed.1.2 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ed.1.23 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ed.2.4 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ed.2.21 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ed.3.3 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ed.3.22 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ed.4.7 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ed.4.18 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ee.1.1 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ee.1.23 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ee.2.2 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ee.2.21 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ef.1.3 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ef.1.21 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ef.2.6 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.15-40.ef.2.17 | $40$ | $2$ | $2$ | $15$ | $0$ | $4^{2}$ |
| 40.576.17-40.mt.1.4 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{8}\cdot2$ |
| 40.576.17-40.mu.1.4 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{8}\cdot2$ |
| 40.576.17-40.mv.1.4 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{8}\cdot2$ |
| 40.576.17-40.mw.1.4 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{8}\cdot2$ |
| 40.576.17-40.mx.1.6 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 40.576.17-40.mx.2.9 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 40.576.17-40.my.1.4 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 40.576.17-40.my.2.6 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 40.1440.43-40.oe.1.10 | $40$ | $5$ | $5$ | $43$ | $5$ | $1^{36}$ |