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Properties

Label 40.72.1.v.1
Level $40$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $20$ Newform level: $20$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ Cusp orbits $2^{6}$
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: 20H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.72.1.143

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}3&16\\24&5\end{bmatrix}$, $\begin{bmatrix}7&20\\24&13\end{bmatrix}$, $\begin{bmatrix}11&8\\32&27\end{bmatrix}$, $\begin{bmatrix}17&39\\36&5\end{bmatrix}$, $\begin{bmatrix}21&36\\0&37\end{bmatrix}$, $\begin{bmatrix}37&10\\20&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.144.1-40.v.1.1, 40.144.1-40.v.1.2, 40.144.1-40.v.1.3, 40.144.1-40.v.1.4, 40.144.1-40.v.1.5, 40.144.1-40.v.1.6, 40.144.1-40.v.1.7, 40.144.1-40.v.1.8, 40.144.1-40.v.1.9, 40.144.1-40.v.1.10, 40.144.1-40.v.1.11, 40.144.1-40.v.1.12, 40.144.1-40.v.1.13, 40.144.1-40.v.1.14, 40.144.1-40.v.1.15, 40.144.1-40.v.1.16, 80.144.1-40.v.1.1, 80.144.1-40.v.1.2, 80.144.1-40.v.1.3, 80.144.1-40.v.1.4, 80.144.1-40.v.1.5, 80.144.1-40.v.1.6, 80.144.1-40.v.1.7, 80.144.1-40.v.1.8, 80.144.1-40.v.1.9, 80.144.1-40.v.1.10, 80.144.1-40.v.1.11, 80.144.1-40.v.1.12, 80.144.1-40.v.1.13, 80.144.1-40.v.1.14, 80.144.1-40.v.1.15, 80.144.1-40.v.1.16, 80.144.1-40.v.1.17, 80.144.1-40.v.1.18, 80.144.1-40.v.1.19, 80.144.1-40.v.1.20, 80.144.1-40.v.1.21, 80.144.1-40.v.1.22, 80.144.1-40.v.1.23, 80.144.1-40.v.1.24, 120.144.1-40.v.1.1, 120.144.1-40.v.1.2, 120.144.1-40.v.1.3, 120.144.1-40.v.1.4, 120.144.1-40.v.1.5, 120.144.1-40.v.1.6, 120.144.1-40.v.1.7, 120.144.1-40.v.1.8, 120.144.1-40.v.1.9, 120.144.1-40.v.1.10, 120.144.1-40.v.1.11, 120.144.1-40.v.1.12, 120.144.1-40.v.1.13, 120.144.1-40.v.1.14, 120.144.1-40.v.1.15, 120.144.1-40.v.1.16, 240.144.1-40.v.1.1, 240.144.1-40.v.1.2, 240.144.1-40.v.1.3, 240.144.1-40.v.1.4, 240.144.1-40.v.1.5, 240.144.1-40.v.1.6, 240.144.1-40.v.1.7, 240.144.1-40.v.1.8, 240.144.1-40.v.1.9, 240.144.1-40.v.1.10, 240.144.1-40.v.1.11, 240.144.1-40.v.1.12, 240.144.1-40.v.1.13, 240.144.1-40.v.1.14, 240.144.1-40.v.1.15, 240.144.1-40.v.1.16, 240.144.1-40.v.1.17, 240.144.1-40.v.1.18, 240.144.1-40.v.1.19, 240.144.1-40.v.1.20, 240.144.1-40.v.1.21, 240.144.1-40.v.1.22, 240.144.1-40.v.1.23, 240.144.1-40.v.1.24, 280.144.1-40.v.1.1, 280.144.1-40.v.1.2, 280.144.1-40.v.1.3, 280.144.1-40.v.1.4, 280.144.1-40.v.1.5, 280.144.1-40.v.1.6, 280.144.1-40.v.1.7, 280.144.1-40.v.1.8, 280.144.1-40.v.1.9, 280.144.1-40.v.1.10, 280.144.1-40.v.1.11, 280.144.1-40.v.1.12, 280.144.1-40.v.1.13, 280.144.1-40.v.1.14, 280.144.1-40.v.1.15, 280.144.1-40.v.1.16
Cyclic 40-isogeny field degree: $2$
Cyclic 40-torsion field degree: $32$
Full 40-torsion field degree: $10240$

Jacobian

Conductor: $2^{2}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 20.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 10 x^{2} + 4 z w - w^{2} $
$=$ $10 x y + 10 y^{2} - 5 z^{2} + z w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{4} - 10 x^{2} y^{2} + 12 x^{2} y z - 4 x^{2} z^{2} + y^{2} z^{2} - 2 y z^{3} + z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{5}w$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{184320000y^{2}z^{16}-217497600000y^{2}z^{15}w+3497299200000y^{2}z^{14}w^{2}-13416624000000y^{2}z^{13}w^{3}+25212260160000y^{2}z^{12}w^{4}-28918771200000y^{2}z^{11}w^{5}+22452072768000y^{2}z^{10}w^{6}-12487926528000y^{2}z^{9}w^{7}+5141176128000y^{2}z^{8}w^{8}-1594566432000y^{2}z^{7}w^{9}+375030864000y^{2}z^{6}w^{10}-66652848000y^{2}z^{5}w^{11}+8821612800y^{2}z^{4}w^{12}-843897600y^{2}z^{3}w^{13}+55200600y^{2}z^{2}w^{14}-2211300y^{2}zw^{15}+40950y^{2}w^{16}-92672000z^{18}+113395200000z^{17}w-2335117440000z^{16}w^{2}+11360098112000z^{15}w^{3}-26757841440000z^{14}w^{4}+38380983552000z^{13}w^{5}-37413801267200z^{12}w^{6}+26364322752000z^{11}w^{7}-13941651916800z^{10}w^{8}+5660756041600z^{9}w^{9}-1787951188800z^{8}w^{10}+441629620800z^{7}w^{11}-85157845840z^{6}w^{12}+12700458000z^{5}w^{13}-1437678060z^{4}w^{14}+119513030z^{3}w^{15}-6885375z^{2}w^{16}+245760zw^{17}-4096w^{18}}{z^{5}(4z-w)^{2}(5z-w)(3200y^{2}z^{8}-46400y^{2}z^{7}w+114400y^{2}z^{6}w^{2}-112800y^{2}z^{5}w^{3}+57600y^{2}z^{4}w^{4}-16800y^{2}z^{3}w^{5}+2840y^{2}z^{2}w^{6}-260y^{2}zw^{7}+10y^{2}w^{8}-1600z^{10}+16480z^{9}w-31216z^{8}w^{2}+24912z^{7}w^{3}-10608z^{6}w^{4}+2624z^{5}w^{5}-380z^{4}w^{6}+30z^{3}w^{7}-z^{2}w^{8})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_0(20)$ $20$ $2$ $2$ $1$ $0$ dimension zero
40.36.0.b.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.36.0.d.1 $40$ $2$ $2$ $0$ $0$ full Jacobian

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.144.5.k.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.be.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.ed.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.ef.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.fl.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.fo.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.gf.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.gh.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.gs.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.gt.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.ha.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.hb.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.hm.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.hn.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.hq.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.hr.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.360.13.bh.1 $40$ $5$ $5$ $13$ $0$ $1^{6}\cdot2^{3}$
80.144.3.i.1 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.i.2 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.j.1 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.j.2 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.k.2 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.k.4 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.l.2 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.l.4 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.o.2 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.p.2 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.s.1 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.t.1 $80$ $2$ $2$ $3$ $?$ not computed
80.144.7.bi.1 $80$ $2$ $2$ $7$ $?$ not computed
80.144.7.bj.1 $80$ $2$ $2$ $7$ $?$ not computed
80.144.7.bm.2 $80$ $2$ $2$ $7$ $?$ not computed
80.144.7.bn.2 $80$ $2$ $2$ $7$ $?$ not computed
120.144.5.bdj.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bdl.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bdx.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bdz.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bpj.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bpl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bpx.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bpz.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzi.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzj.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzq.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzr.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cac.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cad.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cag.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cah.1 $120$ $2$ $2$ $5$ $?$ not computed
120.216.13.eb.1 $120$ $3$ $3$ $13$ $?$ not computed
120.288.13.eoj.1 $120$ $4$ $4$ $13$ $?$ not computed
200.360.13.v.1 $200$ $5$ $5$ $13$ $?$ not computed
240.144.3.i.2 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.i.4 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.j.2 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.j.4 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.k.1 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.k.2 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.l.1 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.l.2 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.o.2 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.p.2 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.s.1 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.t.1 $240$ $2$ $2$ $3$ $?$ not computed
240.144.7.ty.1 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.tz.1 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.uc.2 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.ud.2 $240$ $2$ $2$ $7$ $?$ not computed
280.144.5.qt.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.qu.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.ra.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.rb.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.sx.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.sy.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.te.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.tf.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.uc.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.ud.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.ug.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.uh.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.uk.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.ul.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.uo.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.up.2 $280$ $2$ $2$ $5$ $?$ not computed