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Properties

Label 40.144.3-20.a.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: 40.144.3.777

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}3&30\\30&13\end{bmatrix}$, $\begin{bmatrix}23&18\\12&39\end{bmatrix}$, $\begin{bmatrix}31&20\\12&39\end{bmatrix}$, $\begin{bmatrix}33&10\\16&17\end{bmatrix}$, $\begin{bmatrix}35&12\\28&39\end{bmatrix}$, $\begin{bmatrix}39&0\\2&17\end{bmatrix}$
Contains $-I$: no $\quad$ (see 20.72.3.a.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: $2^{10}\cdot5^{3}$
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 w - z w^{2} $
$=$ $x^{2} w - x y w + w^{2} t$
$=$ $x y z - z^{2} w$
$=$ $x^{2} z - x y z + z w t$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} y - x^{4} z - 5 x^{2} y^{2} z + 6 x^{2} y z^{2} - 2 x^{2} z^{3} + y z^{4} - z^{5} $
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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
$(0:1:0:0:0)$, $(0:0:1:0:1)$, $(0:0:0:0:1)$, $(0:0:0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{400y^{10}t+10400y^{8}t^{3}+140400y^{6}t^{5}+1320800y^{4}t^{7}+9780400y^{2}t^{9}-1843201z^{2}w^{9}-15051201z^{2}w^{8}t-22942730z^{2}w^{7}t^{2}-10567159z^{2}w^{6}t^{3}-1589149z^{2}w^{5}t^{4}+186641z^{2}w^{4}t^{5}-1447520z^{2}w^{3}t^{6}+791014z^{2}w^{2}t^{7}-1995941z^{2}wt^{8}+9780375z^{2}t^{9}-1433602zw^{10}+922800zw^{9}t+41151347zw^{8}t^{2}+59194777zw^{7}t^{3}+24472325zw^{6}t^{4}+3858570zw^{5}t^{5}-2501917zw^{4}t^{6}+4548178zw^{3}t^{7}-8952880zw^{2}t^{8}+13096932zwt^{9}-9780375zt^{10}-w^{11}+5w^{10}t-7168715w^{9}t^{2}-38394720w^{8}t^{3}-43220010w^{7}t^{4}-14712179w^{6}t^{5}-3749385w^{5}t^{6}+5750860w^{4}t^{7}-12312355w^{3}t^{8}+16582225w^{2}t^{9}-48901891wt^{10}}{t^{2}w^{3}(z^{2}w^{4}+7z^{2}w^{3}t+39z^{2}w^{2}t^{2}+112z^{2}wt^{3}+16z^{2}t^{4}+2zw^{5}+12zw^{4}t+59zw^{3}t^{2}-119zw^{2}t^{3}-272zwt^{4}-32zt^{5}+w^{6}+w^{5}t-12w^{4}t^{2}-35w^{3}t^{3}+128w^{2}t^{4}+176wt^{5}+16t^{6})}$

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

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle t$
$\displaystyle Z$ $=$ $\displaystyle z$

Equation of the image curve:

$0$ $=$ $ X^{4}Y-X^{4}Z-5X^{2}Y^{2}Z+6X^{2}YZ^{2}-2X^{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 20.72.3.a.1 :

$\displaystyle X$ $=$ $\displaystyle -y$
$\displaystyle Y$ $=$ $\displaystyle 3y^{2}z^{2}-5y^{2}zt$
$\displaystyle Z$ $=$ $\displaystyle z$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
40.24.0-4.a.1.3 $40$ $6$ $6$ $0$ $0$ full Jacobian
40.72.1-20.b.1.6 $40$ $2$ $2$ $1$ $0$ $1^{2}$
40.72.1-20.b.1.9 $40$ $2$ $2$ $1$ $0$ $1^{2}$

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.a.1.10 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-20.a.2.8 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.b.1.7 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.b.2.7 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-20.c.1.6 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-20.c.2.4 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.h.1.7 $40$ $2$ $2$ $5$ $0$ $2$
40.288.5-40.h.2.7 $40$ $2$ $2$ $5$ $0$ $2$
40.288.7-20.a.1.9 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.7-40.a.1.13 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-20.b.1.17 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.7-40.b.1.13 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-20.c.1.9 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-20.d.1.5 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.7-40.d.1.13 $40$ $2$ $2$ $7$ $4$ $1^{4}$
40.288.7-40.f.1.13 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.7-20.h.1.15 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-20.h.2.11 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-20.i.1.6 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-20.i.2.6 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.m.1.7 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.m.2.1 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.n.1.5 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.n.2.3 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.w.1.11 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.w.2.11 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.y.1.11 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.y.2.11 $40$ $2$ $2$ $7$ $0$ $2^{2}$
40.288.7-40.bk.1.3 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-40.bl.1.1 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.288.7-40.bo.1.5 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.288.7-40.bp.1.7 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.288.9-40.a.1.12 $40$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
40.288.9-40.b.1.13 $40$ $2$ $2$ $9$ $0$ $1^{4}\cdot2$
40.288.9-40.e.1.9 $40$ $2$ $2$ $9$ $3$ $1^{4}\cdot2$
40.288.9-40.f.1.11 $40$ $2$ $2$ $9$ $2$ $1^{4}\cdot2$
40.288.9-40.i.1.13 $40$ $2$ $2$ $9$ $0$ $2\cdot4$
40.288.9-40.i.2.11 $40$ $2$ $2$ $9$ $0$ $2\cdot4$
40.288.9-40.j.1.15 $40$ $2$ $2$ $9$ $0$ $2\cdot4$
40.288.9-40.j.2.9 $40$ $2$ $2$ $9$ $0$ $2\cdot4$
40.720.19-20.a.1.18 $40$ $5$ $5$ $19$ $3$ $1^{16}$
120.288.5-60.bc.1.10 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-60.bc.2.8 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-60.be.1.8 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-60.be.2.8 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.dh.1.15 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.dh.2.11 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.dn.1.13 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.dn.2.11 $120$ $2$ $2$ $5$ $?$ not computed
120.288.7-60.p.1.24 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-60.q.1.10 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-60.y.1.16 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-60.z.1.18 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cp.1.26 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.cr.1.26 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.ee.1.26 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.eg.1.26 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-60.ej.1.9 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-60.ej.2.14 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-60.ek.1.10 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-60.ek.2.14 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.yj.1.7 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.yj.2.14 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.yk.1.5 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.yk.2.13 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bcn.1.21 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bcn.2.17 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bcp.1.23 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bcp.2.19 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bdz.1.3 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bea.1.1 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bed.1.11 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bee.1.9 $120$ $2$ $2$ $7$ $?$ not computed
120.288.9-120.iq.1.19 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.ir.1.29 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.lg.1.17 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.lh.1.25 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.bca.1.19 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.bca.2.23 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.bcb.1.27 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9-120.bcb.2.31 $120$ $2$ $2$ $9$ $?$ not computed
120.432.15-60.a.1.67 $120$ $3$ $3$ $15$ $?$ not computed
280.288.5-140.i.1.10 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-140.i.2.8 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-140.k.1.7 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-140.k.2.6 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.z.1.15 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.z.2.11 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.bf.1.11 $280$ $2$ $2$ $5$ $?$ not computed
280.288.5-280.bf.2.11 $280$ $2$ $2$ $5$ $?$ not computed
280.288.7-140.c.1.13 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.c.1.22 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-140.d.1.17 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.e.1.22 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-140.f.1.13 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-140.g.1.5 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.j.1.22 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.l.1.22 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-140.m.1.11 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-140.m.2.13 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-140.n.1.11 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-140.n.2.11 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bc.1.11 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bc.2.14 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bd.1.9 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bd.2.13 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bm.1.17 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bm.2.19 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bo.1.19 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.bo.2.23 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.ca.1.3 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.cb.1.1 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.ce.1.7 $280$ $2$ $2$ $7$ $?$ not computed
280.288.7-280.cf.1.5 $280$ $2$ $2$ $7$ $?$ not computed
280.288.9-280.a.1.18 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.b.1.27 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.e.1.17 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.f.1.25 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.i.1.21 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.i.2.23 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.j.1.29 $280$ $2$ $2$ $9$ $?$ not computed
280.288.9-280.j.2.31 $280$ $2$ $2$ $9$ $?$ not computed