Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $1600$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $10^{4}\cdot40^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40A8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.8.996 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&2\\12&1\end{bmatrix}$, $\begin{bmatrix}29&23\\28&39\end{bmatrix}$, $\begin{bmatrix}37&12\\36&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.8.cy.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{34}\cdot5^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{8}$ |
| Newforms: | 50.2.a.b$^{2}$, 400.2.a.a, 400.2.a.f, 1600.2.a.j$^{2}$, 1600.2.a.x$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x r + y v + 2 w u $ |
| $=$ | $x t - 2 y u + z v$ | |
| $=$ | $x t - x u + y u - z v + w v + w r$ | |
| $=$ | $2 x v - y v - y r - w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 1024 x^{10} - 1152 x^{8} y^{2} + 2880 x^{8} z^{2} - 260 x^{6} y^{4} + 3380 x^{6} y^{2} z^{2} + \cdots + 250 y^{4} z^{6} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, 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 40.60.4.bh.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle v$ |
| $\displaystyle W$ | $=$ | $\displaystyle -r$ |
Equation of the image curve:
| $0$ | $=$ | $ 70X^{2}+10Y^{2}+Z^{2}-W^{2} $ |
| $=$ | $ 10X^{3}-10XY^{2}-XZ^{2}-YZW $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.120.8.cy.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}t$ |
Equation of the image curve:
| $0$ | $=$ | $ -1024X^{10}-1152X^{8}Y^{2}+2880X^{8}Z^{2}-260X^{6}Y^{4}+3380X^{6}Y^{2}Z^{2}-5225X^{6}Z^{4}-140X^{4}Y^{6}+490X^{4}Y^{4}Z^{2}-2150X^{4}Y^{2}Z^{4}+4500X^{4}Z^{6}-12X^{2}Y^{8}+40X^{2}Y^{6}Z^{2}-325X^{2}Y^{4}Z^{4}+1500X^{2}Y^{2}Z^{6}-2500X^{2}Z^{8}-4Y^{10}+50Y^{8}Z^{2}-200Y^{6}Z^{4}+250Y^{4}Z^{6} $ |
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.by.1.1 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.120.4-40.bh.1.1 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-40.bh.1.5 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-40.bp.1.1 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-40.bp.1.3 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-40.bq.1.9 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-40.bq.1.13 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.720.22-40.go.1.3 | $40$ | $3$ | $3$ | $22$ | $8$ | $1^{14}$ |
| 40.960.29-40.yy.1.1 | $40$ | $4$ | $4$ | $29$ | $5$ | $1^{21}$ |
| 80.480.16-80.cc.1.2 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.cc.2.3 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.cd.1.1 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.16-80.cd.2.1 | $80$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 80.480.17-80.bt.1.16 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.bv.1.11 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.dz.1.15 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.eb.1.9 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.18-80.dy.1.2 | $80$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 80.480.18-80.dy.2.3 | $80$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 80.480.18-80.eb.1.2 | $80$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 80.480.18-80.eb.2.3 | $80$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 240.480.16-240.cs.1.1 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.cs.2.3 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.ct.1.3 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.16-240.ct.2.1 | $240$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 240.480.17-240.eh.1.15 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.ej.1.23 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.kd.1.7 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.kf.1.11 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.18-240.dy.1.1 | $240$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 240.480.18-240.dy.2.1 | $240$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 240.480.18-240.eb.1.1 | $240$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 240.480.18-240.eb.2.1 | $240$ | $2$ | $2$ | $18$ | $?$ | not computed |