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Properties

Label 40.72.1.p.2
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: $80$
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 $2^{2}\cdot4^{2}$
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: 40.72.1.166

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}13&1\\26&13\end{bmatrix}$, $\begin{bmatrix}23&39\\34&23\end{bmatrix}$, $\begin{bmatrix}29&27\\12&9\end{bmatrix}$, $\begin{bmatrix}37&5\\14&13\end{bmatrix}$, $\begin{bmatrix}39&35\\10&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.144.1-40.p.2.1, 40.144.1-40.p.2.2, 40.144.1-40.p.2.3, 40.144.1-40.p.2.4, 40.144.1-40.p.2.5, 40.144.1-40.p.2.6, 40.144.1-40.p.2.7, 40.144.1-40.p.2.8, 80.144.1-40.p.2.1, 80.144.1-40.p.2.2, 80.144.1-40.p.2.3, 80.144.1-40.p.2.4, 80.144.1-40.p.2.5, 80.144.1-40.p.2.6, 80.144.1-40.p.2.7, 80.144.1-40.p.2.8, 80.144.1-40.p.2.9, 80.144.1-40.p.2.10, 80.144.1-40.p.2.11, 80.144.1-40.p.2.12, 80.144.1-40.p.2.13, 80.144.1-40.p.2.14, 80.144.1-40.p.2.15, 80.144.1-40.p.2.16, 120.144.1-40.p.2.1, 120.144.1-40.p.2.2, 120.144.1-40.p.2.3, 120.144.1-40.p.2.4, 120.144.1-40.p.2.5, 120.144.1-40.p.2.6, 120.144.1-40.p.2.7, 120.144.1-40.p.2.8, 240.144.1-40.p.2.1, 240.144.1-40.p.2.2, 240.144.1-40.p.2.3, 240.144.1-40.p.2.4, 240.144.1-40.p.2.5, 240.144.1-40.p.2.6, 240.144.1-40.p.2.7, 240.144.1-40.p.2.8, 240.144.1-40.p.2.9, 240.144.1-40.p.2.10, 240.144.1-40.p.2.11, 240.144.1-40.p.2.12, 240.144.1-40.p.2.13, 240.144.1-40.p.2.14, 240.144.1-40.p.2.15, 240.144.1-40.p.2.16, 280.144.1-40.p.2.1, 280.144.1-40.p.2.2, 280.144.1-40.p.2.3, 280.144.1-40.p.2.4, 280.144.1-40.p.2.5, 280.144.1-40.p.2.6, 280.144.1-40.p.2.7, 280.144.1-40.p.2.8
Cyclic 40-isogeny field degree: $4$
Cyclic 40-torsion field degree: $64$
Full 40-torsion field degree: $10240$

Jacobian

Conductor: $2^{4}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 80.2.a.b

Models

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

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

$ 0 $ $=$ $ x^{4} + 4 x^{2} z^{2} - 2 y^{2} z^{2} + 20 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^6\,\frac{216yz^{17}-432yz^{15}w^{2}-360yz^{13}w^{4}+1768yz^{11}w^{6}-2120yz^{9}w^{8}+1296yz^{7}w^{10}-442yz^{5}w^{12}+80yz^{3}w^{14}-6yzw^{16}+2160z^{18}-11664z^{16}w^{2}+25920z^{14}w^{4}-31036z^{12}w^{6}+21992z^{10}w^{8}-9480z^{8}w^{10}+2416z^{6}w^{12}-320z^{4}w^{14}+12z^{2}w^{16}+w^{18}}{z^{10}(2z^{2}-w^{2})^{2}(2yz^{3}-2yzw^{2}+20z^{4}-13z^{2}w^{2}+2w^{4})}$

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
20.36.1.b.1 $20$ $2$ $2$ $1$ $0$ dimension zero
40.36.0.b.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.36.0.c.2 $40$ $2$ $2$ $0$ $0$ 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.h.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.bc.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.cl.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.cp.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.ej.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.em.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.fd.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.ff.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.360.13.bb.1 $40$ $5$ $5$ $13$ $1$ $1^{6}\cdot2^{3}$
80.144.3.m.1 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.n.1 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.q.2 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.r.2 $80$ $2$ $2$ $3$ $?$ not computed
80.144.7.bg.2 $80$ $2$ $2$ $7$ $?$ not computed
80.144.7.bh.2 $80$ $2$ $2$ $7$ $?$ not computed
80.144.7.bk.1 $80$ $2$ $2$ $7$ $?$ not computed
80.144.7.bl.1 $80$ $2$ $2$ $7$ $?$ not computed
120.144.5.bch.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bcj.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bcv.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bcx.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.boh.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.boj.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bov.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.box.1 $120$ $2$ $2$ $5$ $?$ not computed
120.216.13.dv.2 $120$ $3$ $3$ $13$ $?$ not computed
120.288.13.eod.2 $120$ $4$ $4$ $13$ $?$ not computed
200.360.13.p.2 $200$ $5$ $5$ $13$ $?$ not computed
240.144.3.m.1 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.n.1 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.q.2 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.r.2 $240$ $2$ $2$ $3$ $?$ not computed
240.144.7.tw.2 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.tx.2 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.ua.1 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.ub.1 $240$ $2$ $2$ $7$ $?$ not computed
280.144.5.pr.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.ps.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.py.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.pz.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.rv.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.rw.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.sc.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.sd.1 $280$ $2$ $2$ $5$ $?$ not computed