Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| Index: | $360$ | $\PSL_2$-index: | $180$ | ||||
| Genus: | $10 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $5^{6}\cdot10^{3}\cdot40^{3}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 7$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 7$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-16$) |
Other labels
| Cummins and Pauli (CP) label: | 40D10 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.360.10.274 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&2\\20&17\end{bmatrix}$, $\begin{bmatrix}13&7\\0&9\end{bmatrix}$, $\begin{bmatrix}17&33\\16&33\end{bmatrix}$, $\begin{bmatrix}23&35\\20&33\end{bmatrix}$, $\begin{bmatrix}27&14\\0&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.180.10.cw.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $64$ |
| Full 40-torsion field degree: | $2048$ |
Jacobian
| Conductor: | $2^{30}\cdot5^{17}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}$ |
| Newforms: | 20.2.a.a, 50.2.a.b$^{2}$, 80.2.a.a, 80.2.a.b, 100.2.a.a, 400.2.a.a, 400.2.a.c, 400.2.a.e, 400.2.a.f |
Models
Canonical model in $\mathbb{P}^{ 9 }$ defined by 28 equations
| $ 0 $ | $=$ | $ y^{2} - y v + r a $ |
| $=$ | $y^{2} - y z + u r$ | |
| $=$ | $x z - x v - w r + r a$ | |
| $=$ | $x y + x z - y^{2} + z^{2} + t r - t s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 4 x^{7} y^{2} z^{7} - 8 x^{7} z^{9} + 8 x^{6} y^{6} z^{4} - 28 x^{6} y^{2} z^{8} + x^{5} y^{10} z + \cdots + y^{10} z^{6} $ |
Rational points
This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:0:0:0:-1/2:1)$, $(0:0:0:1:1:0:0:0:0:0)$ |
Maps to other modular curves
Map of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve 40.60.4.bq.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y$ |
| $\displaystyle W$ | $=$ | $\displaystyle s$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}+XZ+2Z^{2}+YW $ |
| $=$ | $ 2X^{2}Z+Y^{2}Z-2Z^{3}-XYW-YZW+2XW^{2}-2ZW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.180.10.cw.1 :
| $\displaystyle X$ | $=$ | $\displaystyle a$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2r$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{12}+X^{5}Y^{10}Z+8X^{3}Y^{12}Z+21X^{4}Y^{10}Z^{2}+14X^{2}Y^{12}Z^{2}+22X^{5}Y^{8}Z^{3}+6X^{3}Y^{10}Z^{3}-8XY^{12}Z^{3}+8X^{6}Y^{6}Z^{4}-5X^{4}Y^{8}Z^{4}-2X^{2}Y^{10}Z^{4}+Y^{12}Z^{4}+3X^{5}Y^{6}Z^{5}+37X^{3}Y^{8}Z^{5}-3XY^{10}Z^{5}+3X^{4}Y^{6}Z^{6}+9X^{2}Y^{8}Z^{6}+Y^{10}Z^{6}-4X^{7}Y^{2}Z^{7}-42X^{5}Y^{4}Z^{7}+X^{3}Y^{6}Z^{7}+XY^{8}Z^{7}-28X^{6}Y^{2}Z^{8}-20X^{4}Y^{4}Z^{8}-3X^{2}Y^{6}Z^{8}-8X^{7}Z^{9}-12X^{5}Y^{2}Z^{9}-2X^{3}Y^{4}Z^{9} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.180.4-20.c.1.3 | $20$ | $2$ | $2$ | $4$ | $0$ | $1^{6}$ |
| 40.120.4-40.bq.1.13 | $40$ | $3$ | $3$ | $4$ | $1$ | $1^{6}$ |
| 40.180.4-20.c.1.19 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.720.19-40.ou.1.19 | $40$ | $2$ | $2$ | $19$ | $3$ | $1^{9}$ |
| 40.720.19-40.ow.1.9 | $40$ | $2$ | $2$ | $19$ | $6$ | $1^{9}$ |
| 40.720.19-40.pc.1.9 | $40$ | $2$ | $2$ | $19$ | $5$ | $1^{9}$ |
| 40.720.19-40.pe.1.9 | $40$ | $2$ | $2$ | $19$ | $7$ | $1^{9}$ |
| 40.720.19-40.qe.1.14 | $40$ | $2$ | $2$ | $19$ | $4$ | $1^{9}$ |
| 40.720.19-40.qg.1.10 | $40$ | $2$ | $2$ | $19$ | $6$ | $1^{9}$ |
| 40.720.19-40.qm.1.9 | $40$ | $2$ | $2$ | $19$ | $3$ | $1^{9}$ |
| 40.720.19-40.qo.1.10 | $40$ | $2$ | $2$ | $19$ | $7$ | $1^{9}$ |
| 40.720.22-40.ch.1.46 | $40$ | $2$ | $2$ | $22$ | $2$ | $1^{12}$ |
| 40.720.22-40.ci.1.10 | $40$ | $2$ | $2$ | $22$ | $4$ | $1^{12}$ |
| 40.720.22-40.fc.1.4 | $40$ | $2$ | $2$ | $22$ | $6$ | $1^{12}$ |
| 40.720.22-40.fe.1.10 | $40$ | $2$ | $2$ | $22$ | $12$ | $1^{12}$ |
| 40.720.22-40.ga.1.10 | $40$ | $2$ | $2$ | $22$ | $2$ | $1^{12}$ |
| 40.720.22-40.gc.1.3 | $40$ | $2$ | $2$ | $22$ | $8$ | $1^{12}$ |
| 40.720.22-40.gm.1.10 | $40$ | $2$ | $2$ | $22$ | $6$ | $1^{12}$ |
| 40.720.22-40.go.1.3 | $40$ | $2$ | $2$ | $22$ | $8$ | $1^{12}$ |
| 40.720.22-40.hs.1.5 | $40$ | $2$ | $2$ | $22$ | $10$ | $1^{12}$ |
| 40.720.22-40.hu.1.5 | $40$ | $2$ | $2$ | $22$ | $6$ | $1^{12}$ |
| 40.720.22-40.ia.1.5 | $40$ | $2$ | $2$ | $22$ | $6$ | $1^{12}$ |
| 40.720.22-40.ic.1.9 | $40$ | $2$ | $2$ | $22$ | $4$ | $1^{12}$ |
| 40.720.22-40.iy.1.10 | $40$ | $2$ | $2$ | $22$ | $4$ | $1^{12}$ |
| 40.720.22-40.ja.1.5 | $40$ | $2$ | $2$ | $22$ | $4$ | $1^{12}$ |
| 40.720.22-40.jg.1.6 | $40$ | $2$ | $2$ | $22$ | $12$ | $1^{12}$ |
| 40.720.22-40.ji.1.3 | $40$ | $2$ | $2$ | $22$ | $4$ | $1^{12}$ |