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Properties

Label 40.72.1.bn.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: $400$
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: 40.72.1.195

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&35\\10&21\end{bmatrix}$, $\begin{bmatrix}3&34\\32&25\end{bmatrix}$, $\begin{bmatrix}5&2\\14&13\end{bmatrix}$, $\begin{bmatrix}27&16\\0&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 80.144.1-40.bn.2.1, 80.144.1-40.bn.2.2, 80.144.1-40.bn.2.3, 80.144.1-40.bn.2.4, 80.144.1-40.bn.2.5, 80.144.1-40.bn.2.6, 80.144.1-40.bn.2.7, 80.144.1-40.bn.2.8, 80.144.1-40.bn.2.9, 80.144.1-40.bn.2.10, 80.144.1-40.bn.2.11, 80.144.1-40.bn.2.12, 80.144.1-40.bn.2.13, 80.144.1-40.bn.2.14, 80.144.1-40.bn.2.15, 80.144.1-40.bn.2.16, 240.144.1-40.bn.2.1, 240.144.1-40.bn.2.2, 240.144.1-40.bn.2.3, 240.144.1-40.bn.2.4, 240.144.1-40.bn.2.5, 240.144.1-40.bn.2.6, 240.144.1-40.bn.2.7, 240.144.1-40.bn.2.8, 240.144.1-40.bn.2.9, 240.144.1-40.bn.2.10, 240.144.1-40.bn.2.11, 240.144.1-40.bn.2.12, 240.144.1-40.bn.2.13, 240.144.1-40.bn.2.14, 240.144.1-40.bn.2.15, 240.144.1-40.bn.2.16
Cyclic 40-isogeny field degree: $4$
Cyclic 40-torsion field degree: $64$
Full 40-torsion field degree: $10240$

Jacobian

Conductor: $2^{4}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 400.2.a.c

Models

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

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

$ 0 $ $=$ $ 5 x^{4} + 4 x^{2} z^{2} - 10 y^{2} z^{2} + 4 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle \frac{5}{2}z$

Maps to other modular curves

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

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

Modular covers

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Cover information

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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.e.1 $20$ $2$ $2$ $1$ $0$ dimension zero
40.36.0.b.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.36.0.c.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.t.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.bk.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.cz.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.dd.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.iq.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.ir.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.ja.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.jd.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.360.13.cd.1 $40$ $5$ $5$ $13$ $1$ $1^{6}\cdot2^{3}$
120.144.5.chj.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.chn.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cil.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cip.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.egy.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ehb.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eia.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eid.1 $120$ $2$ $2$ $5$ $?$ not computed
120.216.13.ul.2 $120$ $3$ $3$ $13$ $?$ not computed
120.288.13.iep.2 $120$ $4$ $4$ $13$ $?$ not computed
200.360.13.bn.2 $200$ $5$ $5$ $13$ $?$ not computed
280.144.5.bfz.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bgb.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bgn.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bgp.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.boq.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bor.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bpe.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bpf.1 $280$ $2$ $2$ $5$ $?$ not computed