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Properties

Label 40.144.3-40.d.1.2
Level $40$
Index $144$
Genus $3$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $40$ $\SL_2$-level: $20$ Newform level: $1600$
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: 40.144.3.221

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}5&24\\2&17\end{bmatrix}$, $\begin{bmatrix}7&4\\10&1\end{bmatrix}$, $\begin{bmatrix}15&6\\26&5\end{bmatrix}$, $\begin{bmatrix}23&34\\16&1\end{bmatrix}$, $\begin{bmatrix}29&4\\30&33\end{bmatrix}$
Contains $-I$: no $\quad$ (see 40.72.3.d.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: $2^{14}\cdot5^{5}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}$
Newforms: 20.2.a.a, 1600.2.a.k, 1600.2.a.w

Models

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

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

$ 0 $ $=$ $ x^{4} y + x^{2} y^{2} z - 40 x^{2} y z^{2} + 500 x^{2} z^{3} + 100 y z^{4} $
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Weierstrass model Weierstrass model

$ y^{2} + x^{4} y $ $=$ $ -20x^{6} - 50x^{4} - 2000x^{2} + 2500 $
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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
$(0:0:0:0:1)$, $(1:0:1:0:0)$, $(0:1:0:0:0)$, $(0:0:1:0:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{70312500000xz^{10}+11256093750000xz^{8}t^{2}-100597367750000xz^{6}t^{4}+46719924455500xz^{4}t^{6}+5757951566760xz^{2}t^{8}-88848484174xt^{10}+31250000000y^{11}+16406250000y^{9}w^{2}-457031250000y^{9}wt+468437500000y^{9}t^{2}+2172031250000y^{7}w^{2}t^{2}-2276355625000y^{7}wt^{3}+7971556625000y^{7}t^{4}+977086500000y^{5}w^{2}t^{4}-2725818760000y^{5}wt^{5}+646715893000y^{5}t^{6}+282203506750y^{3}w^{2}t^{6}-79070898010y^{3}wt^{7}+69027617100y^{3}t^{8}-15051009495yw^{2}t^{8}+268325929195ywt^{9}-214325485772yt^{10}+125000000000z^{11}-485156250000z^{9}w^{2}-2311718750000z^{9}wt-11053750000000z^{9}t^{2}+19197328125000z^{7}w^{2}t^{2}+30769799375000z^{7}wt^{3}+101606649000000z^{7}t^{4}-71636957625000z^{5}w^{2}t^{4}-7134208715000z^{5}wt^{5}-53206603168000z^{5}t^{6}+3046013988000z^{3}w^{2}t^{6}-7763870950290z^{3}wt^{7}-4805034948960z^{3}t^{8}+1153429858657zw^{2}t^{8}-723540728847zwt^{9}+303298969946zt^{10}}{t(13000000000xz^{8}t-1884000000xz^{6}t^{3}-256880000xz^{4}t^{5}+768480xz^{2}t^{7}+48185xt^{9}-625000000y^{9}w+656250000y^{9}t-396875000y^{7}w^{2}t+422500000y^{7}wt^{2}+208500000y^{7}t^{3}+30775000y^{5}w^{2}t^{3}+44785000y^{5}wt^{4}-13790000y^{5}t^{5}+9222800y^{3}w^{2}t^{5}-2612450y^{3}wt^{6}-1636950y^{3}t^{7}-140228yw^{2}t^{7}+376656ywt^{8}-173973yt^{9}-2500000000z^{9}w-13000000000z^{9}t+12350000000z^{7}w^{2}t-960000000z^{7}wt^{2}+2934000000z^{7}t^{3}+1811600000z^{5}w^{2}t^{3}-622160000z^{5}wt^{4}+472480000z^{5}t^{5}+15231200z^{3}w^{2}t^{5}-19812750z^{3}wt^{6}+7527520z^{3}t^{7}+17277zw^{2}t^{7}-280855zwt^{8}+125788zt^{9})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.72.3.d.1 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle t$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{10}w$

Equation of the image curve:

$0$ $=$ $ X^{4}Y+X^{2}Y^{2}Z-40X^{2}YZ^{2}+500X^{2}Z^{3}+100YZ^{4} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 40.72.3.d.1 :

$\displaystyle X$ $=$ $\displaystyle -w$
$\displaystyle Y$ $=$ $\displaystyle -50y^{4}+20y^{2}w^{2}-10y^{2}wt-w^{4}$
$\displaystyle Z$ $=$ $\displaystyle y$

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.1.1 $20$ $2$ $2$ $1$ $0$ $1^{2}$
40.24.0-40.b.1.4 $40$ $6$ $6$ $0$ $0$ full Jacobian
40.72.1-10.a.1.1 $40$ $2$ $2$ $1$ $0$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.288.5-40.p.1.7 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.p.2.5 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.r.1.3 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.r.2.3 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.v.1.1 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.v.2.2 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.x.1.2 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.x.2.1 $40$ $2$ $2$ $5$ $0$ $2$
40.288.7-40.f.1.1 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.7-40.f.1.2 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.7-40.g.1.1 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.288.7-40.g.1.2 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.288.7-40.h.1.1 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-40.h.1.4 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-40.i.1.5 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-40.i.1.14 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-40.bi.1.1 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.bi.1.4 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.bi.2.1 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.bi.2.4 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.bj.1.1 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.bj.1.4 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.bj.2.1 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.bj.2.4 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.720.19-40.p.1.1 $40$ $5$ $5$ $19$ $5$ $1^{16}$
120.288.5-120.et.1.5 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.et.2.6 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.ev.1.7 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.ev.2.5 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.ez.1.5 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.ez.2.6 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.fb.1.7 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.fb.2.5 $120$ $2$ $2$ $5$ $?$ not computed
120.288.7-120.eo.1.25 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.eo.1.29 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.eq.1.17 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.eq.1.23 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.eu.1.1 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.eu.1.23 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.ew.1.13 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.ew.1.25 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bdl.1.4 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bdl.1.11 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bdl.2.4 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bdl.2.7 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bdm.1.2 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bdm.1.14 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bdm.2.2 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bdm.2.12 $120$ $2$ $2$ $7$ $?$ not computed
120.432.15-120.f.1.4 $120$ $3$ $3$ $15$ $?$ not computed
280.288.5-280.bn.1.5 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.bn.2.6 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.bp.1.7 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.bp.2.5 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.bt.1.5 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.bt.2.6 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.bv.1.6 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.bv.2.5 $280$ $2$ $2$ $5$ $?$ not computed
280.288.7-280.x.1.2 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.x.1.6 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.y.1.2 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.y.1.6 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.ba.1.2 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.ba.1.14 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bb.1.2 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bb.1.14 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.by.1.3 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.by.1.8 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.by.2.3 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.by.2.8 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bz.1.2 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bz.1.8 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bz.2.2 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bz.2.8 $280$ $2$ $2$ $7$ $?$ not computed