Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $400$ | ||
| 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: | 40.240.8.385 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&12\\1&29\end{bmatrix}$, $\begin{bmatrix}31&36\\29&11\end{bmatrix}$, $\begin{bmatrix}35&4\\39&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.8.cw.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: | $2^{26}\cdot5^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{8}$ |
| Newforms: | 50.2.a.b$^{2}$, 400.2.a.a, 400.2.a.d$^{2}$, 400.2.a.f, 400.2.a.h$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x t - x u - y u - z r + w v - w r $ |
| $=$ | $x t + 2 y u + z v$ | |
| $=$ | $2 x v + y v - y r - z t - w t$ | |
| $=$ | $2 x v - y v - y r + z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 256 x^{10} + 288 x^{8} y^{2} + 1440 x^{8} z^{2} + 65 x^{6} y^{4} + 1690 x^{6} y^{2} z^{2} + \cdots + 500 y^{4} z^{6} $ |
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 20.60.4.l.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -v$ |
| $\displaystyle W$ | $=$ | $\displaystyle r$ |
Equation of the image curve:
| $0$ | $=$ | $ 35X^{2}+5Y^{2}-Z^{2}+W^{2} $ |
| $=$ | $ 5X^{3}-5XY^{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.cw.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x-y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}t$ |
Equation of the image curve:
| $0$ | $=$ | $ 256X^{10}+288X^{8}Y^{2}+1440X^{8}Z^{2}+65X^{6}Y^{4}+1690X^{6}Y^{2}Z^{2}+5225X^{6}Z^{4}+35X^{4}Y^{6}+245X^{4}Y^{4}Z^{2}+2150X^{4}Y^{2}Z^{4}+9000X^{4}Z^{6}+3X^{2}Y^{8}+20X^{2}Y^{6}Z^{2}+325X^{2}Y^{4}Z^{4}+3000X^{2}Y^{2}Z^{6}+10000X^{2}Z^{8}+Y^{10}+25Y^{8}Z^{2}+200Y^{6}Z^{4}+500Y^{4}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.120.4-20.l.1.2 | $20$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.48.0-40.bw.1.2 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.120.4-20.l.1.6 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-40.bo.1.3 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-40.bo.1.5 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 40.120.4-40.bq.1.11 | $40$ | $2$ | $2$ | $4$ | $1$ | $1^{4}$ |
| 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.gm.1.10 | $40$ | $3$ | $3$ | $22$ | $6$ | $1^{14}$ |
| 40.960.29-40.yw.1.4 | $40$ | $4$ | $4$ | $29$ | $7$ | $1^{21}$ |
| 80.480.17-80.bs.1.8 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.bu.1.8 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.dy.1.7 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 80.480.17-80.ea.1.7 | $80$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.eg.1.15 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.ei.1.16 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.kc.1.7 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 240.480.17-240.ke.1.8 | $240$ | $2$ | $2$ | $17$ | $?$ | not computed |