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Properties

Label 40.72.1.bn.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: $400$
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: 40.72.1.196

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}9&13\\28&19\end{bmatrix}$, $\begin{bmatrix}19&33\\10&17\end{bmatrix}$, $\begin{bmatrix}21&23\\12&17\end{bmatrix}$, $\begin{bmatrix}31&0\\38&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 80.144.1-40.bn.1.1, 80.144.1-40.bn.1.2, 80.144.1-40.bn.1.3, 80.144.1-40.bn.1.4, 80.144.1-40.bn.1.5, 80.144.1-40.bn.1.6, 80.144.1-40.bn.1.7, 80.144.1-40.bn.1.8, 80.144.1-40.bn.1.9, 80.144.1-40.bn.1.10, 80.144.1-40.bn.1.11, 80.144.1-40.bn.1.12, 80.144.1-40.bn.1.13, 80.144.1-40.bn.1.14, 80.144.1-40.bn.1.15, 80.144.1-40.bn.1.16, 240.144.1-40.bn.1.1, 240.144.1-40.bn.1.2, 240.144.1-40.bn.1.3, 240.144.1-40.bn.1.4, 240.144.1-40.bn.1.5, 240.144.1-40.bn.1.6, 240.144.1-40.bn.1.7, 240.144.1-40.bn.1.8, 240.144.1-40.bn.1.9, 240.144.1-40.bn.1.10, 240.144.1-40.bn.1.11, 240.144.1-40.bn.1.12, 240.144.1-40.bn.1.13, 240.144.1-40.bn.1.14, 240.144.1-40.bn.1.15, 240.144.1-40.bn.1.16
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 $ $=$ $ 2 x^{2} + z^{2} + z w $
$=$ $4 x^{2} + 10 y^{2} - 2 z^{2} - 6 z w - 5 w^{2}$
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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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{(16z^{6}+80z^{5}w+160z^{4}w^{2}+160z^{3}w^{3}+80z^{2}w^{4}+20zw^{5}+5w^{6})^{3}}{w^{5}z^{2}(z+w)^{10}(4z+5w)}$

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