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: | $6$ (of which $2$ are rational) | Cusp widths | $20^{6}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20A8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.8.663 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&2\\36&21\end{bmatrix}$, $\begin{bmatrix}23&22\\28&33\end{bmatrix}$, $\begin{bmatrix}25&38\\34&25\end{bmatrix}$, $\begin{bmatrix}27&38\\28&13\end{bmatrix}$, $\begin{bmatrix}29&26\\16&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.8.c.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: | $2^{30}\cdot5^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{8}$ |
| Newforms: | 50.2.a.b$^{3}$, 200.2.a.e, 1600.2.a.a, 1600.2.a.i, 1600.2.a.q, 1600.2.a.y |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x t + x r + y r + z v $ |
| $=$ | $2 x r - y t + w v$ | |
| $=$ | $x u + x v + y u - z r - w t$ | |
| $=$ | $2 x w + 2 y z + u r + v r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 32 x^{10} - 16 x^{8} y^{2} + 32 x^{8} z^{2} - 2 x^{6} y^{4} + 44 x^{6} y^{2} z^{2} + \cdots + 8 y^{2} z^{8} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:1:-1:1:0)$, $(0:0:0:0:-1:-1:1:0)$ |
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.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
| $\displaystyle W$ | $=$ | $\displaystyle -w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}+XY+2Y^{2}-Z^{2}-ZW $ |
| $=$ | $ 2X^{2}Y+2XY^{2}+2XZW+YZW+XW^{2}+YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.120.8.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}t$ |
Equation of the image curve:
| $0$ | $=$ | $ -32X^{10}-16X^{8}Y^{2}+32X^{8}Z^{2}-2X^{6}Y^{4}+44X^{6}Y^{2}Z^{2}-8X^{6}Z^{4}-X^{4}Y^{4}Z^{2}+2X^{4}Y^{2}Z^{4}-6X^{2}Y^{6}Z^{2}-8X^{2}Y^{4}Z^{4}-24X^{2}Y^{2}Z^{6}-Y^{8}Z^{2}+6Y^{6}Z^{4}-12Y^{4}Z^{6}+8Y^{2}Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.0-8.c.1.7 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.120.4-20.b.1.15 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-20.b.1.18 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.480.16-40.g.1.4 | $40$ | $2$ | $2$ | $16$ | $6$ | $2^{4}$ |
| 40.480.16-40.g.1.13 | $40$ | $2$ | $2$ | $16$ | $6$ | $2^{4}$ |
| 40.480.16-40.h.1.8 | $40$ | $2$ | $2$ | $16$ | $2$ | $2^{4}$ |
| 40.480.16-40.h.1.9 | $40$ | $2$ | $2$ | $16$ | $2$ | $2^{4}$ |
| 40.480.16-40.i.1.8 | $40$ | $2$ | $2$ | $16$ | $2$ | $2^{4}$ |
| 40.480.16-40.i.1.15 | $40$ | $2$ | $2$ | $16$ | $2$ | $2^{4}$ |
| 40.480.16-40.i.2.7 | $40$ | $2$ | $2$ | $16$ | $2$ | $2^{4}$ |
| 40.480.16-40.i.2.16 | $40$ | $2$ | $2$ | $16$ | $2$ | $2^{4}$ |
| 40.480.16-40.j.1.6 | $40$ | $2$ | $2$ | $16$ | $4$ | $2^{4}$ |
| 40.480.16-40.j.1.16 | $40$ | $2$ | $2$ | $16$ | $4$ | $2^{4}$ |
| 40.480.16-40.j.2.6 | $40$ | $2$ | $2$ | $16$ | $4$ | $2^{4}$ |
| 40.480.16-40.j.2.16 | $40$ | $2$ | $2$ | $16$ | $4$ | $2^{4}$ |
| 40.480.16-40.k.1.4 | $40$ | $2$ | $2$ | $16$ | $2$ | $2^{4}$ |
| 40.480.16-40.k.1.13 | $40$ | $2$ | $2$ | $16$ | $2$ | $2^{4}$ |
| 40.480.16-40.l.1.2 | $40$ | $2$ | $2$ | $16$ | $2$ | $2^{4}$ |
| 40.480.16-40.l.1.15 | $40$ | $2$ | $2$ | $16$ | $2$ | $2^{4}$ |
| 40.480.17-40.bd.1.6 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{7}\cdot2$ |
| 40.480.17-40.bd.1.12 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{7}\cdot2$ |
| 40.480.17-40.bi.1.3 | $40$ | $2$ | $2$ | $17$ | $6$ | $1^{7}\cdot2$ |
| 40.480.17-40.bi.1.16 | $40$ | $2$ | $2$ | $17$ | $6$ | $1^{7}\cdot2$ |
| 40.480.17-40.br.1.7 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{7}\cdot2$ |
| 40.480.17-40.br.1.16 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{7}\cdot2$ |
| 40.480.17-40.bt.1.8 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{7}\cdot2$ |
| 40.480.17-40.bt.1.14 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{7}\cdot2$ |
| 40.720.22-40.c.1.31 | $40$ | $3$ | $3$ | $22$ | $6$ | $1^{14}$ |
| 40.960.29-40.t.1.16 | $40$ | $4$ | $4$ | $29$ | $9$ | $1^{21}$ |
| 120.480.16-120.g.1.11 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.g.1.29 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.h.1.7 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.h.1.27 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.i.1.3 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.i.1.29 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.i.2.7 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.i.2.27 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.j.1.4 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.j.1.21 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.j.2.11 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.j.2.20 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.k.1.8 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.k.1.11 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.l.1.3 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.l.1.16 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.17-120.bo.1.4 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.bo.1.26 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.br.1.12 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.br.1.18 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.ca.1.14 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.ca.1.18 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.cd.1.6 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.cd.1.26 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.16-280.g.1.9 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.g.1.31 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.h.1.11 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.h.1.23 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.i.1.11 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.i.1.21 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.i.2.11 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.i.2.23 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.j.1.5 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.j.1.20 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.j.2.11 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.j.2.20 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.k.1.11 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.k.1.20 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.l.1.3 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.l.1.24 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.17-280.bo.1.2 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.bo.1.30 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.br.1.4 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.br.1.26 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ca.1.10 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ca.1.20 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.cd.1.2 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.cd.1.24 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |