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: | $6$ (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: | $1$ | ||||||
| $\Q$-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: | 40.240.8.444 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&18\\34&9\end{bmatrix}$, $\begin{bmatrix}3&2\\0&9\end{bmatrix}$, $\begin{bmatrix}3&8\\4&7\end{bmatrix}$, $\begin{bmatrix}7&2\\34&3\end{bmatrix}$, $\begin{bmatrix}27&38\\30&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.120.8.e.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: | $2^{18}\cdot5^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{8}$ |
| Newforms: | 50.2.a.a, 50.2.a.b$^{3}$, 200.2.a.a, 200.2.a.e, 400.2.a.d, 400.2.a.h |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ y z + y w - y t + y u + 2 y r - z v - w t - w u $ |
| $=$ | $x z + x t - x u + y r - z t - z u - z v - w t - w u$ | |
| $=$ | $x y + x t + x u - y v - z t + z u + z r - w t + w u$ | |
| $=$ | $x z + 2 x w - x t + x u + 2 x r + y z + y w + y t - y u + w t + w u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 825 x^{5} y^{6} - 400 x^{5} y^{5} z - 1500 x^{5} y^{4} z^{2} - 3000 x^{5} y^{3} z^{3} + \cdots + 8 y z^{10} $ |
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 |
|---|
| $(0:0:0:0:1:0:-1:1)$, $(0:0:0:0:0:-1:1:1)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x+y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
| $\displaystyle W$ | $=$ | $\displaystyle w$ |
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 20.120.8.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y+z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ -825X^{5}Y^{6}-400X^{5}Y^{5}Z-1500X^{5}Y^{4}Z^{2}-3000X^{5}Y^{3}Z^{3}-3500X^{5}Y^{2}Z^{4}-2400X^{5}YZ^{5}-800X^{5}Z^{6}-1500X^{4}Y^{7}+2000X^{4}Y^{6}Z+2500X^{4}Y^{5}Z^{2}-2500X^{4}Y^{3}Z^{4}-3000X^{4}Y^{2}Z^{5}-1000X^{4}YZ^{6}-1500X^{3}Y^{8}+2000X^{3}Y^{7}Z+2500X^{3}Y^{6}Z^{2}-2500X^{3}Y^{4}Z^{4}-3000X^{3}Y^{3}Z^{5}-1000X^{3}Y^{2}Z^{6}-750X^{2}Y^{9}+1000X^{2}Y^{8}Z+1250X^{2}Y^{7}Z^{2}-1250X^{2}Y^{5}Z^{4}-1500X^{2}Y^{4}Z^{5}-500X^{2}Y^{3}Z^{6}-150XY^{10}+350XY^{9}Z+200XY^{8}Z^{2}-450XY^{7}Z^{3}-500XY^{6}Z^{4}-50XY^{5}Z^{5}+450XY^{4}Z^{6}+400XY^{3}Z^{7}+100XY^{2}Z^{8}-28Y^{11}+20Y^{10}Z+120Y^{9}Z^{2}+60Y^{8}Z^{3}-280Y^{7}Z^{4}-252Y^{6}Z^{5}+140Y^{5}Z^{6}+240Y^{4}Z^{7}+120Y^{3}Z^{8}+40Y^{2}Z^{9}+8YZ^{10} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.48.0-20.c.1.6 | $40$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.120.4-20.b.1.4 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 40.120.4-20.b.1.6 | $40$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
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.o.1.6 | $40$ | $2$ | $2$ | $16$ | $3$ | $2^{4}$ |
| 40.480.16-40.o.1.14 | $40$ | $2$ | $2$ | $16$ | $3$ | $2^{4}$ |
| 40.480.16-40.o.2.9 | $40$ | $2$ | $2$ | $16$ | $3$ | $2^{4}$ |
| 40.480.16-40.o.2.11 | $40$ | $2$ | $2$ | $16$ | $3$ | $2^{4}$ |
| 40.480.16-40.p.1.3 | $40$ | $2$ | $2$ | $16$ | $3$ | $2^{4}$ |
| 40.480.16-40.p.1.4 | $40$ | $2$ | $2$ | $16$ | $3$ | $2^{4}$ |
| 40.480.16-40.p.2.5 | $40$ | $2$ | $2$ | $16$ | $3$ | $2^{4}$ |
| 40.480.16-40.p.2.7 | $40$ | $2$ | $2$ | $16$ | $3$ | $2^{4}$ |
| 40.480.16-40.q.1.3 | $40$ | $2$ | $2$ | $16$ | $1$ | $2^{4}$ |
| 40.480.16-40.q.1.4 | $40$ | $2$ | $2$ | $16$ | $1$ | $2^{4}$ |
| 40.480.16-40.q.2.5 | $40$ | $2$ | $2$ | $16$ | $1$ | $2^{4}$ |
| 40.480.16-40.q.2.7 | $40$ | $2$ | $2$ | $16$ | $1$ | $2^{4}$ |
| 40.480.16-40.r.1.5 | $40$ | $2$ | $2$ | $16$ | $1$ | $2^{4}$ |
| 40.480.16-40.r.1.13 | $40$ | $2$ | $2$ | $16$ | $1$ | $2^{4}$ |
| 40.480.16-40.r.2.13 | $40$ | $2$ | $2$ | $16$ | $1$ | $2^{4}$ |
| 40.480.16-40.r.2.14 | $40$ | $2$ | $2$ | $16$ | $1$ | $2^{4}$ |
| 40.480.17-40.z.1.13 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{7}\cdot2$ |
| 40.480.17-40.z.1.15 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{7}\cdot2$ |
| 40.480.17-40.bh.1.11 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{7}\cdot2$ |
| 40.480.17-40.bh.1.15 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{7}\cdot2$ |
| 40.480.17-40.cq.1.11 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{7}\cdot2$ |
| 40.480.17-40.cq.1.15 | $40$ | $2$ | $2$ | $17$ | $5$ | $1^{7}\cdot2$ |
| 40.480.17-40.cs.1.13 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{7}\cdot2$ |
| 40.480.17-40.cs.1.15 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{7}\cdot2$ |
| 40.720.22-20.e.1.7 | $40$ | $3$ | $3$ | $22$ | $2$ | $1^{14}$ |
| 40.960.29-20.r.1.11 | $40$ | $4$ | $4$ | $29$ | $4$ | $1^{21}$ |
| 120.480.16-120.bg.1.12 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bg.1.18 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bg.2.5 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bg.2.30 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bh.1.15 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bh.1.30 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bh.2.14 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bh.2.23 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bi.1.16 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bi.1.23 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bi.2.8 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bi.2.27 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bj.1.14 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bj.1.18 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bj.2.9 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bj.2.28 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.17-120.cv.1.17 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.cv.1.31 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.cy.1.23 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.cy.1.25 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.hg.1.15 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.hg.1.19 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.hk.1.7 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.hk.1.27 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.16-280.o.1.10 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.o.1.20 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.o.2.5 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.o.2.30 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.p.1.12 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.p.1.23 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.p.2.14 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.p.2.24 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.q.1.10 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.q.1.24 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.q.2.16 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.q.2.20 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.r.1.10 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.r.1.22 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.r.2.9 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.r.2.28 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.17-280.cu.1.7 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.cu.1.27 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.cw.1.11 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.cw.1.23 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ea.1.11 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ea.1.23 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ec.1.3 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ec.1.31 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |