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: | $3$ | ||||||
| $\Q$-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: | 40.240.8.1022 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&8\\36&21\end{bmatrix}$, $\begin{bmatrix}15&4\\11&21\end{bmatrix}$, $\begin{bmatrix}27&32\\25&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.8.x.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.d, 400.2.a.h, 1600.2.a.b, 1600.2.a.i, 1600.2.a.p, 1600.2.a.y |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x u - x r + y r - z v + w t + t v $ |
| $=$ | $x u - 2 y r - z w$ | |
| $=$ | $x u + 3 x r + y u - y r + w t + t v$ | |
| $=$ | $2 x z + 2 x t + 2 y z - 2 y t + w r - v r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{12} + 32 x^{10} y^{2} + 24 x^{8} y^{4} + 180 x^{8} y^{2} z^{2} + 8 x^{6} y^{6} + \cdots + 2500 y^{4} z^{8} $ |
Rational points
This modular curve has no real points, 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.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
| $\displaystyle W$ | $=$ | $\displaystyle -v$ |
Equation of the image curve:
| $0$ | $=$ | $ 14X^{2}+2Y^{2}-Z^{2}+W^{2} $ |
| $=$ | $ 2X^{3}-2XY^{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.x.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{10}r$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{12}+32X^{10}Y^{2}+24X^{8}Y^{4}+8X^{6}Y^{6}+X^{4}Y^{8}+180X^{8}Y^{2}Z^{2}+220X^{6}Y^{4}Z^{2}+85X^{4}Y^{6}Z^{2}+10X^{2}Y^{8}Z^{2}-400X^{6}Y^{2}Z^{4}+200X^{4}Y^{4}Z^{4}+100X^{2}Y^{6}Z^{4}+2250X^{2}Y^{4}Z^{6}+500Y^{6}Z^{6}+2500Y^{4}Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.0-40.n.1.1 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.120.4-40.f.1.1 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-40.f.1.5 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-40.bm.1.1 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{4}$ |
| 40.120.4-40.bm.1.8 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{4}$ |
| 40.120.4-40.bo.1.1 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-40.bo.1.14 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
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.cj.1.3 | $40$ | $3$ | $3$ | $22$ | $6$ | $1^{14}$ |
| 40.960.29-40.hz.1.1 | $40$ | $4$ | $4$ | $29$ | $11$ | $1^{21}$ |