Invariants
| Level: | $40$ | $\SL_2$-level: | $40$ | 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: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20H3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.144.3.1538 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}7&37\\24&15\end{bmatrix}$, $\begin{bmatrix}9&35\\8&21\end{bmatrix}$, $\begin{bmatrix}23&5\\30&3\end{bmatrix}$, $\begin{bmatrix}37&27\\10&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.3.ci.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $4$ |
| Cyclic 40-torsion field degree: | $64$ |
| Full 40-torsion field degree: | $5120$ |
Jacobian
| Conductor: | $2^{16}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 80.2.a.b, 320.2.c.b |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - y u + w t $ |
| $=$ | $x u + z w$ | |
| $=$ | $x t + y z$ | |
| $=$ | $13 x^{2} + 6 x y + y^{2} + 4 z^{2} - z t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{4} + 2 x^{4} y^{2} z^{2} + x^{4} z^{4} - 76 x^{2} y^{4} z^{2} - 244 x^{2} y^{2} z^{4} + \cdots + 5120 z^{8} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -16x^{8} + 64x^{6} - 88x^{4} + 80x^{2} - 25 $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^8}\cdot\frac{2131432704xw^{9}-17459608320xw^{7}u^{2}-36508343328xw^{5}u^{4}-12587882112xw^{3}u^{6}+2982519063xwu^{8}+1859334912yw^{9}-589545216yw^{7}u^{2}-8176160736yw^{5}u^{4}-4437125568yw^{3}u^{6}+382329153ywu^{8}+2000000000zt^{8}u-12354000000zt^{6}u^{3}-6581250000zt^{4}u^{5}-2257584750zt^{2}u^{7}-1270872828zu^{9}+2000000000t^{9}u+718000000t^{7}u^{3}+3486150000t^{5}u^{5}+313823250t^{3}u^{7}+260888175tu^{9}}{12032xw^{9}+24896xw^{7}u^{2}-2064xw^{5}u^{4}-716xw^{3}u^{6}-89xwu^{8}+10496yw^{9}+2240yw^{7}u^{2}-1392yw^{5}u^{4}+108yw^{3}u^{6}-31ywu^{8}-120000zt^{4}u^{5}+13050zt^{2}u^{7}-204zu^{9}+40000t^{5}u^{5}-7350t^{3}u^{7}+335tu^{9}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.3.ci.1 :
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{10}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{20}u$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{4}+2X^{4}Y^{2}Z^{2}-76X^{2}Y^{4}Z^{2}+X^{4}Z^{4}-244X^{2}Y^{2}Z^{4}+3380Y^{4}Z^{4}-168X^{2}Z^{6}+8320Y^{2}Z^{6}+5120Z^{8} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 40.72.3.ci.1 :
| $\displaystyle X$ | $=$ | $\displaystyle \frac{20}{33}w^{2}t^{4}-\frac{19}{165}w^{2}t^{2}u^{2}+\frac{169}{13200}w^{2}u^{4}+\frac{5}{33}t^{4}u^{2}-\frac{83}{1320}t^{2}u^{4}+\frac{1}{40}tu^{5}+\frac{13}{3300}u^{6}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{76}{22275}w^{3}t^{10}u^{11}-\frac{4}{22275}w^{3}t^{9}u^{12}+\frac{329}{185625}w^{3}t^{8}u^{13}-\frac{113}{371250}w^{3}t^{7}u^{14}-\frac{63631}{222750000}w^{3}t^{6}u^{15}+\frac{13481}{222750000}w^{3}t^{5}u^{16}+\frac{52897}{2227500000}w^{3}t^{4}u^{17}-\frac{63713}{8910000000}w^{3}t^{3}u^{18}-\frac{19}{22275}wt^{10}u^{13}-\frac{1}{22275}wt^{9}u^{14}+\frac{1949}{4455000}wt^{8}u^{15}-\frac{103}{2227500}wt^{7}u^{16}-\frac{94471}{891000000}wt^{6}u^{17}+\frac{27127}{891000000}wt^{5}u^{18}+\frac{157313}{35640000000}wt^{4}u^{19}-\frac{4901}{2227500000}wt^{3}u^{20}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -\frac{40}{33}w^{2}t^{4}+\frac{38}{165}w^{2}t^{2}u^{2}-\frac{169}{6600}w^{2}u^{4}-\frac{10}{33}t^{4}u^{2}+\frac{13}{220}t^{2}u^{4}+\frac{1}{600}tu^{5}-\frac{13}{1650}u^{6}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1-20.b.1.9 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 40.72.1-20.b.1.10 | $40$ | $2$ | $2$ | $1$ | $0$ | $2$ |
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.h.1.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.z.2.3 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.288.5-40.ce.2.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.ch.2.8 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.ej.1.3 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.el.1.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.ew.1.2 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.ex.1.8 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.720.19-40.qq.1.7 | $40$ | $5$ | $5$ | $19$ | $1$ | $1^{6}\cdot2^{5}$ |
| 80.288.9-80.u.1.12 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.u.1.16 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.v.1.12 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.v.1.16 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.w.1.12 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.w.1.16 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.x.1.12 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.x.1.16 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.y.1.14 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.y.1.16 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.z.1.15 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.z.1.16 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.ba.1.15 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.ba.1.16 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.bb.1.14 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.288.9-80.bb.1.16 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.5-120.bjs.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bjt.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bkg.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bkh.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqe.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqf.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqs.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.bqt.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.15-120.ma.2.28 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 240.288.9-240.bma.1.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bma.1.14 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmb.1.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmb.1.14 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmc.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmc.1.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmd.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmd.1.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bme.1.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bme.1.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmf.1.7 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmf.1.8 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmg.1.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmg.1.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmh.1.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.288.9-240.bmh.1.12 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.288.5-280.xc.1.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.xd.2.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.xj.2.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.xk.2.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.zg.1.5 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.zh.1.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.zn.1.3 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.zo.1.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |