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: | 40.240.8.960 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&19\\36&31\end{bmatrix}$, $\begin{bmatrix}13&5\\20&27\end{bmatrix}$, $\begin{bmatrix}15&17\\16&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.8.cz.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $96$ |
| 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.a, 1600.2.a.j, 1600.2.a.q, 1600.2.a.x |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ 2 x v + y v - y r - z t + w t $ |
| $=$ | $2 x v - y v - y r + z t + z u$ | |
| $=$ | $x r - y v - z u + w t + 2 w u$ | |
| $=$ | $2 x u + 2 y t + 2 y u - z r - w v + w r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 1024 x^{10} - 576 x^{8} y^{2} + 5760 x^{8} z^{2} + 209 x^{6} y^{4} - 3380 x^{6} y^{2} z^{2} + \cdots + 12500 z^{10} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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.cz.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}-576X^{8}Y^{2}+5760X^{8}Z^{2}+209X^{6}Y^{4}-3380X^{6}Y^{2}Z^{2}+6500X^{6}Z^{4}-36X^{4}Y^{6}+430X^{4}Y^{4}Z^{2}-2450X^{4}Y^{2}Z^{4}+17500X^{4}Z^{6}+4X^{2}Y^{8}-60X^{2}Y^{6}Z^{2}+325X^{2}Y^{4}Z^{4}-1000X^{2}Y^{2}Z^{6}+7500X^{2}Z^{8}-50Y^{6}Z^{4}+1000Y^{4}Z^{6}-6250Y^{2}Z^{8}+12500Z^{10} $ |
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.bz.1.7 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.120.4-40.bh.1.1 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-40.bh.1.7 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-40.bo.1.3 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-40.bo.1.7 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-40.br.1.13 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{4}$ |
| 40.120.4-40.br.1.15 | $40$ | $2$ | $2$ | $4$ | $2$ | $1^{4}$ |
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.gp.1.7 | $40$ | $3$ | $3$ | $22$ | $10$ | $1^{14}$ |
| 40.960.29-40.yz.1.4 | $40$ | $4$ | $4$ | $29$ | $9$ | $1^{21}$ |