Invariants
| Level: | $40$ | $\SL_2$-level: | $20$ | Newform level: | $320$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20I3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.366 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&6\\14&29\end{bmatrix}$, $\begin{bmatrix}7&30\\0&27\end{bmatrix}$, $\begin{bmatrix}25&14\\26&23\end{bmatrix}$, $\begin{bmatrix}27&26\\38&35\end{bmatrix}$, $\begin{bmatrix}33&28\\10&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.3.f.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $32$ |
| Full 40-torsion field degree: | $5120$ |
Jacobian
| Conductor: | $2^{14}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 20.2.a.a, 320.2.c.c |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x t^{2} + y w t - z w t + z t^{2} $ |
| $=$ | $x y t + y^{2} w - y z w + y z t$ | |
| $=$ | $x w t + y w^{2} - z w^{2} + z w t$ | |
| $=$ | $x z t + y z w - z^{2} w + z^{2} t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 20 x^{4} y - 20 x^{4} z - 2 x^{2} y^{2} z - 4 x^{2} y z^{2} + 4 x^{2} z^{3} + y z^{4} - z^{5} $ |
Weierstrass model Weierstrass model
| $ y^{2} + x^{4} y $ | $=$ | $ -4x^{6} + 22x^{4} - 80x^{2} + 100 $ |
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.
| Embedded model |
|---|
| $(0:0:0:0:1)$, $(-1:1:1:0:0)$, $(0:1:0:0:0)$, $(0:0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2\,\frac{6414336xw^{10}+2089984xw^{9}t+68413952xw^{8}t^{2}-263903488xw^{7}t^{3}+524740736xw^{6}t^{4}-652434240xw^{5}t^{5}+556092896xw^{4}t^{6}-329870376xw^{3}t^{7}+134833216xw^{2}t^{8}-36812206xwt^{9}+5760420xt^{10}-156250000y^{11}+15625000y^{9}t^{2}-41562500y^{7}t^{4}+4150000y^{5}t^{6}-3734375y^{3}t^{8}+156250000yz^{10}+462500000yz^{8}t^{2}-283562500yz^{6}t^{4}+117545000yz^{4}t^{6}-10218050yz^{2}t^{8}+366488yt^{10}+68750000z^{9}t^{2}+85125000z^{7}t^{4}-21870000z^{5}t^{6}+23429800z^{3}t^{8}+37158912zw^{10}-137961472zw^{9}t+394718720zw^{8}t^{2}-785480576zw^{7}t^{3}+1085307840zw^{6}t^{4}-1084981120zw^{5}t^{5}+778167096zw^{4}t^{6}-406380928zw^{3}t^{7}+148040086zw^{2}t^{8}-37630680zwt^{9}+5652444zt^{10}}{7424xw^{10}+2176xw^{9}t-150400xw^{8}t^{2}+432480xw^{7}t^{3}-566112xw^{6}t^{4}+390840xw^{5}t^{5}-121736xw^{4}t^{6}-9782xw^{3}t^{7}+20157xw^{2}t^{8}-5984xwt^{9}+571xt^{10}-125000y^{7}t^{4}+12500y^{5}t^{6}+3350y^{3}t^{8}+125000yz^{6}t^{4}+7500yz^{4}t^{6}-7700yz^{2}t^{8}-356yt^{10}+55000z^{5}t^{6}+8600z^{3}t^{8}+43008zw^{10}-159808zw^{9}t+157824zw^{8}t^{2}+127760zw^{7}t^{3}-424400zw^{6}t^{4}+399764zw^{5}t^{5}-172336zw^{4}t^{6}+17631zw^{3}t^{7}+15679zw^{2}t^{8}-8125zwt^{9}+1283zt^{10}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.3.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 20X^{4}Y-20X^{4}Z-2X^{2}Y^{2}Z-4X^{2}YZ^{2}+4X^{2}Z^{3}+YZ^{4}-Z^{5} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 40.72.3.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 10x^{4}-2x^{2}w^{2}-2x^{2}wt$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.72.1-10.a.1.3 | $20$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 40.72.1-10.a.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.288.5-40.a.2.6 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.b.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.j.1.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.k.1.8 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.m.1.5 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.288.5-40.n.1.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.v.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.w.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.7-40.y.2.4 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.y.2.7 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.z.2.6 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.z.2.10 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.ba.2.8 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.ba.2.19 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.bb.2.3 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bb.2.12 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bf.1.10 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bf.1.13 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bg.1.11 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{2}\cdot2$ |
| 40.288.7-40.bg.1.13 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{2}\cdot2$ |
| 40.288.7-40.bh.2.3 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.bh.2.9 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.bi.2.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.bi.2.10 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.720.19-40.bx.1.8 | $40$ | $5$ | $5$ | $19$ | $0$ | $1^{6}\cdot2^{5}$ |
| 120.288.5-120.ds.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dt.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eb.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ec.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fc.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fd.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fl.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fm.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.7-120.bcx.2.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcx.2.25 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcy.2.14 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcy.2.25 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdd.1.18 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdd.1.25 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bde.1.17 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bde.1.26 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdq.2.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdq.2.29 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdr.2.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdr.2.29 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdw.2.4 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdw.2.18 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdx.2.3 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdx.2.20 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.432.15-120.x.2.1 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 280.288.5-280.bz.2.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.ca.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cf.1.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cg.1.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cl.1.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cm.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cr.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cs.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.7-280.co.2.12 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.co.2.25 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cp.2.12 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cp.2.19 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cr.2.8 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cr.2.11 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cs.2.7 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cs.2.14 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.da.2.16 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.da.2.25 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.db.1.22 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.db.1.25 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.dd.2.7 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.dd.2.11 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.de.2.5 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.de.2.14 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |