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: | 40.144.3.437 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&32\\38&19\end{bmatrix}$, $\begin{bmatrix}11&20\\4&37\end{bmatrix}$, $\begin{bmatrix}29&20\\26&23\end{bmatrix}$, $\begin{bmatrix}29&34\\30&23\end{bmatrix}$, $\begin{bmatrix}33&36\\24&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.72.3.f.2 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: | $2^{14}\cdot5^{3}$ |
| 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 y t + y^{2} w + y^{2} t - y z w $ |
| $=$ | $x z t + y z w + y z t - z^{2} w$ | |
| $=$ | $x t^{2} + y w t + y t^{2} - z w t$ | |
| $=$ | $x w t + y w^{2} + y w t - z w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 8 x^{5} z + 4 x^{4} y^{2} + 26 x^{4} z^{2} + 16 x^{3} y^{2} z + 44 x^{3} z^{3} + 20 x^{2} y^{4} + \cdots + 4 z^{6} $ |
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 |
|---|
| $(0:0:0:-1:1)$, $(-1:1:1:0:0)$, $(0:1:0:0:0)$, $(0:0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2\,\frac{7387136xw^{10}+69568000xw^{9}t+360804352xw^{8}t^{2}+930249984xw^{7}t^{3}+1421495424xw^{6}t^{4}+1316064832xw^{5}t^{5}+651680480xw^{4}t^{6}+15132328xw^{3}t^{7}-176291784xw^{2}t^{8}-88926258xwt^{9}-12631510xt^{10}-100000y^{11}+45000y^{9}t^{2}-663500y^{7}t^{4}+299000y^{5}t^{6}-1483825y^{3}t^{8}+550000yz^{10}-5960000yz^{8}t^{2}-1073500yz^{6}t^{4}-9990200yz^{4}t^{6}-12130110yz^{2}t^{8}+659990yt^{10}+800000z^{11}+3690000z^{9}t^{2}+5927000z^{7}t^{4}+15655200z^{5}t^{6}+7652160z^{3}t^{8}+38656512zw^{10}+211456000zw^{9}t+665848320zw^{8}t^{2}+1279038592zw^{7}t^{3}+1530357952zw^{6}t^{4}+1038841728zw^{5}t^{5}+188287304zw^{4}t^{6}-316922432zw^{3}t^{7}-281723246zw^{2}t^{8}-76834148zwt^{9}+4438130zt^{10}}{7424xw^{10}+68992xw^{9}t+126848xw^{8}t^{2}-4768xw^{7}t^{3}-206624xw^{6}t^{4}-208488xw^{5}t^{5}-82312xw^{4}t^{6}-15526xw^{3}t^{7}-5167xw^{2}t^{8}-2179xwt^{9}-246xt^{10}-2000y^{7}t^{4}+900y^{5}t^{6}+1370y^{3}t^{8}-14000yz^{6}t^{4}-13700yz^{4}t^{6}+710yz^{2}t^{8}-640yt^{10}+16000z^{7}t^{4}+26800z^{5}t^{6}-980z^{3}t^{8}+43008zw^{10}+227264zw^{9}t+384640zw^{8}t^{2}+143312zw^{7}t^{3}-301424zw^{6}t^{4}-437020zw^{5}t^{5}-273120zw^{4}t^{6}-111373zw^{3}t^{7}-30771zw^{2}t^{8}-3138zwt^{9}-116zt^{10}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.3.f.2 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}+4X^{4}Y^{2}+20X^{2}Y^{4}+8X^{5}Z+16X^{3}Y^{2}Z+26X^{4}Z^{2}+28X^{2}Y^{2}Z^{2}+44X^{3}Z^{3}+24XY^{2}Z^{3}+41X^{2}Z^{4}+20XZ^{5}+4Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 40.72.3.f.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{4}z^{3}w^{3}+\frac{3}{10}zw^{5}+\frac{3}{5}zw^{4}t+\frac{11}{40}zw^{3}t^{2}-\frac{1}{40}zw^{2}t^{3}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{2}{125}z^{2}w^{22}-\frac{18}{125}z^{2}w^{21}t-\frac{72}{125}z^{2}w^{20}t^{2}-\frac{166}{125}z^{2}w^{19}t^{3}-\frac{939}{500}z^{2}w^{18}t^{4}-\frac{741}{500}z^{2}w^{17}t^{5}-\frac{121}{1000}z^{2}w^{16}t^{6}+\frac{633}{500}z^{2}w^{15}t^{7}+\frac{28383}{16000}z^{2}w^{14}t^{8}+\frac{22363}{16000}z^{2}w^{13}t^{9}+\frac{2379}{3200}z^{2}w^{12}t^{10}+\frac{4443}{16000}z^{2}w^{11}t^{11}+\frac{1157}{16000}z^{2}w^{10}t^{12}+\frac{201}{16000}z^{2}w^{9}t^{13}+\frac{21}{16000}z^{2}w^{8}t^{14}+\frac{1}{16000}z^{2}w^{7}t^{15}+\frac{2}{625}w^{24}+\frac{32}{625}w^{23}t+\frac{229}{625}w^{22}t^{2}+\frac{988}{625}w^{21}t^{3}+\frac{2323}{500}w^{20}t^{4}+\frac{12411}{1250}w^{19}t^{5}+\frac{4011}{250}w^{18}t^{6}+\frac{100453}{5000}w^{17}t^{7}+\frac{1583657}{80000}w^{16}t^{8}+\frac{61921}{4000}w^{15}t^{9}+\frac{1542071}{160000}w^{14}t^{10}+\frac{381477}{80000}w^{13}t^{11}+\frac{148911}{80000}w^{12}t^{12}+\frac{45241}{80000}w^{11}t^{13}+\frac{523}{4000}w^{10}t^{14}+\frac{71}{3200}w^{9}t^{15}+\frac{13}{5000}w^{8}t^{16}+\frac{3}{16000}w^{7}t^{17}+\frac{1}{160000}w^{6}t^{18}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w^{6}+\frac{3}{5}w^{5}t+\frac{13}{20}w^{4}t^{2}+\frac{3}{10}w^{3}t^{3}+\frac{1}{20}w^{2}t^{4}$ |
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.1.7 | $20$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 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.1.18 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.b.1.9 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.j.2.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.k.2.21 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.m.2.17 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.288.5-40.n.2.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.288.5-40.v.1.8 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.5-40.w.1.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.288.7-40.y.1.4 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.z.1.2 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.z.1.7 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.ba.1.6 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.ba.1.16 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.bb.1.4 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bb.1.10 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bf.1.4 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bf.1.10 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 40.288.7-40.bg.2.1 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{2}\cdot2$ |
| 40.288.7-40.bg.2.8 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{2}\cdot2$ |
| 40.288.7-40.bh.1.4 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.bh.1.5 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.bi.1.4 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.bi.1.6 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 40.720.19-40.bx.1.6 | $40$ | $5$ | $5$ | $19$ | $0$ | $1^{6}\cdot2^{5}$ |
| 120.288.5-120.ds.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dt.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eb.2.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ec.2.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fc.2.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fd.2.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fl.1.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fm.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.7-120.bcx.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcx.1.10 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcy.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcy.1.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdd.2.5 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdd.2.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bde.2.6 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bde.2.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdq.1.6 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdq.1.20 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdr.1.5 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdr.1.20 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdw.1.4 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdw.1.13 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdx.1.4 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bdx.1.14 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.432.15-120.x.1.54 | $120$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 280.288.5-280.bz.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.ca.1.11 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cf.2.10 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cg.2.11 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cl.2.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cm.2.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cr.1.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.cs.1.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.7-280.co.1.8 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.co.1.10 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cp.1.3 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cp.1.14 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cr.1.10 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cr.1.19 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cs.1.12 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.cs.1.18 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.da.1.6 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.da.1.20 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.db.2.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.db.2.16 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.dd.1.7 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.dd.1.18 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.de.1.6 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.de.1.20 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |