Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | Newform level: | $200$ | ||
| 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 $4$ are rational) | Cusp widths | $10^{2}\cdot20^{3}\cdot40$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40D8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.240.8.103 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&28\\4&15\end{bmatrix}$, $\begin{bmatrix}9&4\\14&1\end{bmatrix}$, $\begin{bmatrix}9&24\\4&21\end{bmatrix}$, $\begin{bmatrix}25&16\\16&37\end{bmatrix}$, $\begin{bmatrix}31&24\\20&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.120.8.n.2 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: | $1^{4}\cdot2^{2}$ |
| Newforms: | 50.2.a.b$^{3}$, 200.2.a.e, 200.2.d.a, 200.2.d.c |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x z + x u - y u - y r - t v $ |
| $=$ | $y z - y u - w v - t v$ | |
| $=$ | $x y - y^{2} - y w + z v$ | |
| $=$ | $2 x t - z v + v r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 64 x^{12} - 880 x^{10} y z + 1848 x^{10} z^{2} + 320 x^{8} y^{3} z - 880 x^{8} y^{2} z^{2} + \cdots - 4 z^{12} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:-1:1:-1:0:-1:2:1)$, $(0:1:1:1:0:-1:2:1)$, $(0:0:0:0:0:1:0:0)$, $(0:0:-1:0:0:-1:0:1)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
| $\displaystyle W$ | $=$ | $\displaystyle t$ |
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.n.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle r$ |
| $\displaystyle Z$ | $=$ | $\displaystyle v$ |
Equation of the image curve:
| $0$ | $=$ | $ 64X^{12}-880X^{10}YZ+320X^{8}Y^{3}Z-16X^{6}Y^{5}Z+1848X^{10}Z^{2}-880X^{8}Y^{2}Z^{2}+280X^{6}Y^{4}Z^{2}+1600X^{8}YZ^{3}-1160X^{6}Y^{3}Z^{3}+40X^{4}Y^{5}Z^{3}-1548X^{8}Z^{4}+1880X^{6}Y^{2}Z^{4}-360X^{4}Y^{4}Z^{4}-1540X^{6}YZ^{5}+1040X^{4}Y^{3}Z^{5}-24X^{2}Y^{5}Z^{5}+626X^{6}Z^{6}-1340X^{4}Y^{2}Z^{6}+150X^{2}Y^{4}Z^{6}+820X^{4}YZ^{7}-350X^{2}Y^{3}Z^{7}+4Y^{5}Z^{7}-201X^{4}Z^{8}+390X^{2}Y^{2}Z^{8}-20Y^{4}Z^{8}-210X^{2}YZ^{9}+40Y^{3}Z^{9}+44X^{2}Z^{10}-40Y^{2}Z^{10}+20YZ^{11}-4Z^{12} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.e.1.15 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.e.1.15 | $8$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 40.120.4-20.b.1.10 | $40$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 40.120.4-20.b.1.17 | $40$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
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.d.2.5 | $40$ | $2$ | $2$ | $16$ | $3$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.f.1.11 | $40$ | $2$ | $2$ | $16$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.j.2.8 | $40$ | $2$ | $2$ | $16$ | $4$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.l.1.7 | $40$ | $2$ | $2$ | $16$ | $2$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.q.2.11 | $40$ | $2$ | $2$ | $16$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.r.2.5 | $40$ | $2$ | $2$ | $16$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.u.2.6 | $40$ | $2$ | $2$ | $16$ | $2$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.v.2.14 | $40$ | $2$ | $2$ | $16$ | $2$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.y.2.13 | $40$ | $2$ | $2$ | $16$ | $2$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.z.2.13 | $40$ | $2$ | $2$ | $16$ | $2$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.bc.2.9 | $40$ | $2$ | $2$ | $16$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.bd.2.14 | $40$ | $2$ | $2$ | $16$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.bh.2.14 | $40$ | $2$ | $2$ | $16$ | $4$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.bj.1.13 | $40$ | $2$ | $2$ | $16$ | $2$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.bn.2.9 | $40$ | $2$ | $2$ | $16$ | $3$ | $1^{4}\cdot2^{2}$ |
| 40.480.16-40.bp.2.10 | $40$ | $2$ | $2$ | $16$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.480.17-40.bb.2.2 | $40$ | $2$ | $2$ | $17$ | $3$ | $1^{3}\cdot2^{3}$ |
| 40.480.17-40.bf.2.7 | $40$ | $2$ | $2$ | $17$ | $1$ | $1^{3}\cdot2^{3}$ |
| 40.480.17-40.bm.2.1 | $40$ | $2$ | $2$ | $17$ | $4$ | $1^{3}\cdot2^{3}$ |
| 40.480.17-40.bo.1.2 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{3}\cdot2^{3}$ |
| 40.480.17-40.by.2.3 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{3}\cdot2^{3}$ |
| 40.480.17-40.bz.2.4 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{3}\cdot2^{3}$ |
| 40.480.17-40.cc.2.2 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{3}\cdot2^{3}$ |
| 40.480.17-40.cd.2.3 | $40$ | $2$ | $2$ | $17$ | $2$ | $1^{3}\cdot2^{3}$ |
| 40.720.22-40.z.2.32 | $40$ | $3$ | $3$ | $22$ | $0$ | $1^{6}\cdot4^{2}$ |
| 40.960.29-40.ep.2.15 | $40$ | $4$ | $4$ | $29$ | $1$ | $1^{9}\cdot2^{2}\cdot4^{2}$ |
| 120.480.16-120.r.2.10 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.t.1.28 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.z.2.22 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bb.2.19 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bt.2.16 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.bv.2.6 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.cb.2.4 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.cd.2.14 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.cj.2.6 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.cl.2.28 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.cr.2.28 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.ct.2.10 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.dh.2.31 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.dj.2.20 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.dp.2.23 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.16-120.dr.2.21 | $120$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 120.480.17-120.dh.2.4 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.dj.2.18 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.dp.2.4 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.dr.2.25 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.gb.2.25 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.gd.2.4 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.gj.2.18 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.480.17-120.gl.2.17 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.16-280.bb.2.11 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.bd.2.26 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.bj.2.22 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.bl.2.12 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.br.2.30 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.bt.2.6 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.bz.2.4 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.cb.2.26 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.ch.2.4 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.cj.2.32 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.cp.2.32 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.cr.2.6 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.cx.2.31 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.cz.2.19 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.df.2.21 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.16-280.dh.2.19 | $280$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 280.480.17-280.en.2.4 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ep.2.18 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ev.2.6 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ex.2.25 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.fd.2.25 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.ff.2.4 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.fl.2.18 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 280.480.17-280.fn.2.17 | $280$ | $2$ | $2$ | $17$ | $?$ | not computed |