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Properties

Label 40.144.3-20.l.1.11
Level $40$
Index $144$
Genus $3$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $40$ $\SL_2$-level: $40$ Newform level: $80$
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: 20J3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.144.3.50

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&27\\12&11\end{bmatrix}$, $\begin{bmatrix}9&20\\32&17\end{bmatrix}$, $\begin{bmatrix}11&3\\8&1\end{bmatrix}$, $\begin{bmatrix}21&33\\4&35\end{bmatrix}$, $\begin{bmatrix}23&18\\12&19\end{bmatrix}$, $\begin{bmatrix}23&36\\4&35\end{bmatrix}$
Contains $-I$: no $\quad$ (see 20.72.3.l.1 for the level structure with $-I$)
Cyclic 40-isogeny field degree: $2$
Cyclic 40-torsion field degree: $16$
Full 40-torsion field degree: $5120$

Jacobian

Conductor: $2^{10}\cdot5^{3}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}$
Newforms: 20.2.a.a, 80.2.a.a, 80.2.a.b

Models

Embedded model Embedded model in $\mathbb{P}^{4}$

$ 0 $ $=$ $ x y z + x y t - y^{2} t + z^{2} t + z t^{2} $
$=$ $x^{2} z + x^{2} t - x y z + x y w - y^{2} t - z^{2} t + z w t - z t^{2} + w t^{2}$
$=$ $2 x y z - x y w - 2 z w t - 2 z t^{2} - w t^{2}$
$=$ $ - x y t + 2 y^{2} z + y^{2} w + y^{2} t + z^{2} t + z w t + z t^{2} + w t^{2}$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{6} - x^{4} y^{2} - 10 x^{4} z^{2} - 8 x^{3} y z^{2} - 2 x^{2} y^{2} z^{2} + x^{2} z^{4} - y^{2} z^{4} $
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Weierstrass model Weierstrass model

$ y^{2} + \left(x^{4} + 1\right) y $ $=$ $ -2x^{6} - x^{4} - 2x^{2} $
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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
$(-4:1:0:0:0)$, $(0:0:-1:-2:1)$, $(0:0:0:0:1)$, $(0:0:-1/2:1:0)$

Maps to other modular curves

$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2^2\,\frac{65536xw^{10}+655360xw^{9}t+3539648xw^{8}t^{2}+12566144xw^{7}t^{3}+30962608xw^{6}t^{4}+53272224xw^{5}t^{5}+59535640xw^{4}t^{6}+33403468xw^{3}t^{7}+2995714xw^{2}t^{8}-3183349xwt^{9}+14797851xt^{10}-20480y^{9}t^{2}+957440y^{7}t^{4}-15869440y^{5}t^{6}+107420340y^{3}t^{8}-260608yzw^{9}-2668160yzw^{8}t-13474368yzw^{7}t^{2}-45371968yzw^{6}t^{3}-109402192yzw^{5}t^{4}-165763128yzw^{4}t^{5}-120247540yzw^{3}t^{6}-21608222yzw^{2}t^{7}+16896632yzwt^{8}+40911688yzt^{9}-130368yw^{10}-1586688yw^{9}t-9486544yw^{8}t^{2}-36709312yw^{7}t^{3}-102676968yw^{6}t^{4}-210192916yw^{5}t^{5}-278892078yw^{4}t^{6}-184704381yw^{3}t^{7}-20666609yw^{2}t^{8}+46610200ywt^{9}+59170924yt^{10}}{t^{2}(32xw^{6}t^{2}-1856xw^{5}t^{3}+4600xw^{4}t^{4}-356xw^{3}t^{5}+8722xw^{2}t^{6}+4679xwt^{7}+2295xt^{8}-5120y^{7}t^{2}+640y^{5}t^{4}+260y^{3}t^{6}+64yzw^{7}-4800yzw^{6}t+21680yzw^{5}t^{2}-5944yzw^{4}t^{3}+50780yzw^{3}t^{4}+20586yzw^{2}t^{5}+27576yzwt^{6}+8744yzt^{7}+32yw^{8}-1728yw^{7}t-3848yw^{6}t^{2}+27900yw^{5}t^{3}-1214yw^{4}t^{4}+75503yw^{3}t^{5}+30299yw^{2}t^{6}+35896ywt^{7}+9180yt^{8})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 20.72.3.l.1 :

$\displaystyle X$ $=$ $\displaystyle z+t$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}w+t$
$\displaystyle Z$ $=$ $\displaystyle y$

Equation of the image curve:

$0$ $=$ $ X^{6}-X^{4}Y^{2}-10X^{4}Z^{2}-8X^{3}YZ^{2}-2X^{2}Y^{2}Z^{2}+X^{2}Z^{4}-Y^{2}Z^{4} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 20.72.3.l.1 :

$\displaystyle X$ $=$ $\displaystyle -\frac{9}{128}yz-\frac{1}{256}yw-\frac{25}{256}yt-\frac{9}{64}z^{2}-\frac{1}{128}zw-\frac{43}{128}zt-\frac{1}{128}wt-\frac{25}{128}t^{2}$
$\displaystyle Y$ $=$ $\displaystyle -\frac{1}{2048}yz^{7}-\frac{1}{4096}yz^{6}w-\frac{157}{32768}yz^{6}t-\frac{119}{65536}yz^{5}wt-\frac{162521}{8388608}yz^{5}t^{2}-\frac{93289}{16777216}yz^{4}wt^{2}-\frac{11336529}{268435456}yz^{4}t^{3}-\frac{4824693}{536870912}yz^{3}wt^{3}-\frac{28954865}{536870912}yz^{3}t^{4}-\frac{4345955}{536870912}yz^{2}wt^{4}-\frac{10884075}{268435456}yz^{2}t^{5}-\frac{2071225}{536870912}yzwt^{5}-\frac{8957625}{536870912}yzt^{6}-\frac{102125}{134217728}ywt^{6}-\frac{390625}{134217728}yt^{7}-\frac{1}{4096}z^{8}-\frac{1}{8192}z^{7}w-\frac{723}{262144}z^{7}t-\frac{531}{524288}z^{6}wt-\frac{215559}{16777216}z^{6}t^{2}-\frac{117783}{33554432}z^{5}wt^{2}-\frac{35219861}{1073741824}z^{5}t^{3}-\frac{14169525}{2147483648}z^{4}wt^{3}-\frac{217964741}{4294967296}z^{4}t^{4}-\frac{31219975}{4294967296}z^{3}wt^{4}-\frac{105506285}{2147483648}z^{3}t^{5}-\frac{20149685}{4294967296}z^{2}wt^{5}-\frac{125718225}{4294967296}z^{2}t^{6}-\frac{880525}{536870912}zwt^{6}-\frac{5303375}{536870912}zt^{7}-\frac{64125}{268435456}wt^{7}-\frac{390625}{268435456}t^{8}$
$\displaystyle Z$ $=$ $\displaystyle -\frac{9}{64}yz-\frac{1}{128}yw-\frac{25}{128}yt-\frac{1}{32}z^{2}-\frac{1}{64}zw-\frac{13}{256}zt-\frac{1}{64}wt$

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
$X_0(5)$ $5$ $24$ $12$ $0$ $0$ full Jacobian
8.24.0-4.d.1.3 $8$ $6$ $6$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0-4.d.1.3 $8$ $6$ $6$ $0$ $0$ full Jacobian
40.72.1-20.c.1.6 $40$ $2$ $2$ $1$ $0$ $1^{2}$
40.72.1-20.c.1.21 $40$ $2$ $2$ $1$ $0$ $1^{2}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.288.5-20.n.1.10 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-20.n.2.6 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-20.p.1.3 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-20.p.2.4 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.dp.1.6 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.dp.2.5 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.ed.1.6 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.ed.2.5 $40$ $2$ $2$ $5$ $0$ $2$
40.288.7-20.p.1.9 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.7-20.q.1.5 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.288.7-20.r.1.9 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-20.r.2.5 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.ck.1.8 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-40.cl.1.16 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.7-40.cq.1.7 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.7-40.cr.1.7 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.7-40.cy.1.4 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.cy.2.3 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.cz.1.2 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.cz.2.1 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.dc.1.7 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-40.dd.1.13 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-40.de.1.10 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.de.2.10 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.dn.1.1 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.dn.2.2 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.do.1.7 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.do.2.8 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.dt.1.2 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-40.du.1.2 $40$ $2$ $2$ $7$ $4$ $1^{4}$
40.288.7-40.dx.1.4 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.7-40.dy.1.4 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.9-40.bs.1.12 $40$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
40.288.9-40.bt.1.12 $40$ $2$ $2$ $9$ $0$ $1^{4}\cdot2$
40.288.9-40.bu.1.12 $40$ $2$ $2$ $9$ $3$ $1^{4}\cdot2$
40.288.9-40.bv.1.14 $40$ $2$ $2$ $9$ $2$ $1^{4}\cdot2$
40.288.9-40.bw.1.12 $40$ $2$ $2$ $9$ $0$ $2\cdot4$
40.288.9-40.bw.2.12 $40$ $2$ $2$ $9$ $0$ $2\cdot4$
40.288.9-40.bx.1.15 $40$ $2$ $2$ $9$ $0$ $2\cdot4$
40.288.9-40.bx.2.14 $40$ $2$ $2$ $9$ $0$ $2\cdot4$
40.720.19-20.bh.1.18 $40$ $5$ $5$ $19$ $3$ $1^{16}$
120.288.5-60.df.1.12 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-60.df.2.10 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-60.dl.1.12 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-60.dl.2.10 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.wl.1.15 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.wl.2.13 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.yb.1.12 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.yb.2.11 $120$ $2$ $2$ $5$ $?$ not computed
120.288.7-60.fp.1.12 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-60.fq.1.22 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-60.fr.1.14 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-60.fr.2.18 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cbh.1.5 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cbi.1.9 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.ceu.1.7 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cev.1.13 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cki.1.15 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cki.2.13 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.ckj.1.13 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.ckj.2.9 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.ckm.1.6 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.ckn.1.10 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cko.1.26 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cko.2.23 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cog.1.16 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cog.2.15 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.coh.1.15 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.coh.2.13 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.com.1.7 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.con.1.13 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.coq.1.8 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cor.1.15 $120$ $2$ $2$ $7$ $?$ not computed
120.288.9-120.czc.1.12 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.czd.1.15 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.cze.1.14 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.czf.1.15 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.czg.1.15 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.czg.2.14 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.czh.1.15 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.czh.2.15 $120$ $2$ $2$ $9$ $?$ not computed
120.432.15-60.bv.1.66 $120$ $3$ $3$ $15$ $?$ not computed
280.288.5-140.bp.1.12 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-140.bp.2.10 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-140.bv.1.8 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-140.bv.2.7 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.ld.1.15 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.ld.2.13 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.mt.1.12 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.mt.2.11 $280$ $2$ $2$ $5$ $?$ not computed
280.288.7-140.bc.1.22 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-140.bd.1.20 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-140.be.1.20 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-140.be.2.16 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.ec.1.5 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.ed.1.9 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.eg.1.7 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.eh.1.13 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.em.1.15 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.em.2.13 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.en.1.13 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.en.2.9 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.eq.1.29 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.er.1.25 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.es.1.22 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.es.2.22 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.fb.1.16 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.fb.2.15 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.fc.1.15 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.fc.2.13 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.fh.1.7 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.fi.1.13 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.fl.1.8 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.fm.1.15 $280$ $2$ $2$ $7$ $?$ not computed
280.288.9-280.bs.1.12 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.bt.1.15 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.bu.1.14 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.bv.1.15 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.bw.1.15 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.bw.2.14 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.bx.1.15 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.bx.2.15 $280$ $2$ $2$ $9$ $?$ not computed