Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $320$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $4^{2}\cdot8^{2}\cdot20^{2}\cdot40^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40J9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.288.9.12 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&34\\4&3\end{bmatrix}$, $\begin{bmatrix}9&24\\28&5\end{bmatrix}$, $\begin{bmatrix}13&1\\12&27\end{bmatrix}$, $\begin{bmatrix}21&25\\28&3\end{bmatrix}$, $\begin{bmatrix}23&5\\4&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.144.9.bt.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: | $2^{46}\cdot5^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}\cdot2$ |
| Newforms: | 20.2.a.a, 64.2.a.a$^{2}$, 80.2.a.a, 80.2.a.b, 320.2.a.b, 320.2.a.e, 320.2.a.g |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ y s + v^{2} $ |
| $=$ | $u s - v r$ | |
| $=$ | $y r + u v$ | |
| $=$ | $y r + t s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 400 x^{4} y^{8} z^{4} - 200 x^{4} y^{4} z^{8} + 25 x^{4} z^{12} - 544 x^{2} y^{12} z^{2} + \cdots + y^{8} z^{8} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:1:-1:1:0:0:0:0:0)$, $(0:-1:-1:1:0:0:0:0:0)$, $(2/3:-5/3:5/3:1:0:0:0:0:0)$, $(2/3:5/3:5/3:1:0:0:0:0:0)$ |
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 40.72.5.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
| $\displaystyle W$ | $=$ | $\displaystyle -u$ |
| $\displaystyle T$ | $=$ | $\displaystyle -v$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}+XY+XZ-W^{2} $ |
| $=$ | $ 2X^{2}-XY+XZ+YZ-Z^{2} $ | |
| $=$ | $ 2XW+YW-3ZW+T^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.9.bt.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle s$ |
Equation of the image curve:
| $0$ | $=$ | $ 144Y^{16}-544X^{2}Y^{12}Z^{2}+400X^{4}Y^{8}Z^{4}-24Y^{12}Z^{4}+224X^{2}Y^{8}Z^{6}-200X^{4}Y^{4}Z^{8}+Y^{8}Z^{8}-26X^{2}Y^{4}Z^{10}+25X^{4}Z^{12}+X^{2}Z^{14} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 8.48.1-8.n.1.4 | $8$ | $6$ | $6$ | $1$ | $0$ | $1^{6}\cdot2$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1-8.n.1.4 | $8$ | $6$ | $6$ | $1$ | $0$ | $1^{6}\cdot2$ |
| 20.144.3-20.l.1.5 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 40.144.3-20.l.1.9 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.576.17-40.mn.1.4 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 40.576.17-40.mu.1.8 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 40.576.17-40.qe.1.3 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{8}$ |
| 40.576.17-40.qk.1.3 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 40.576.17-40.rn.1.15 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 40.576.17-40.rn.2.13 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 40.576.17-40.rv.1.8 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 40.576.17-40.rv.2.7 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 40.576.17-40.si.1.2 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 40.576.17-40.si.2.1 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 40.576.17-40.sq.1.4 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 40.576.17-40.sq.2.3 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 40.576.17-40.to.1.5 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 40.576.17-40.to.2.6 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 40.576.17-40.tw.1.1 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 40.576.17-40.tw.2.2 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 40.576.17-40.ut.1.5 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 40.576.17-40.ut.2.6 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 40.576.17-40.ux.1.5 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 40.576.17-40.ux.2.6 | $40$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 40.576.17-40.vk.1.6 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 40.576.17-40.vq.1.4 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{8}$ |
| 40.576.17-40.wh.1.8 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 40.576.17-40.wm.1.8 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{8}$ |
| 40.1440.49-40.bbl.1.7 | $40$ | $5$ | $5$ | $49$ | $11$ | $1^{28}\cdot2^{6}$ |