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Properties

Label 40.72.1.b.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: $10$ 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 $2^{6}\cdot10^{6}$ 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: 10K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.72.1.55

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}7&30\\38&29\end{bmatrix}$, $\begin{bmatrix}9&38\\32&5\end{bmatrix}$, $\begin{bmatrix}11&28\\10&29\end{bmatrix}$, $\begin{bmatrix}17&0\\28&9\end{bmatrix}$, $\begin{bmatrix}19&20\\16&33\end{bmatrix}$, $\begin{bmatrix}35&12\\16&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.144.1-40.b.1.1, 40.144.1-40.b.1.2, 40.144.1-40.b.1.3, 40.144.1-40.b.1.4, 40.144.1-40.b.1.5, 40.144.1-40.b.1.6, 40.144.1-40.b.1.7, 40.144.1-40.b.1.8, 80.144.1-40.b.1.1, 80.144.1-40.b.1.2, 80.144.1-40.b.1.3, 80.144.1-40.b.1.4, 80.144.1-40.b.1.5, 80.144.1-40.b.1.6, 80.144.1-40.b.1.7, 80.144.1-40.b.1.8, 80.144.1-40.b.1.9, 80.144.1-40.b.1.10, 80.144.1-40.b.1.11, 80.144.1-40.b.1.12, 80.144.1-40.b.1.13, 80.144.1-40.b.1.14, 80.144.1-40.b.1.15, 80.144.1-40.b.1.16, 120.144.1-40.b.1.1, 120.144.1-40.b.1.2, 120.144.1-40.b.1.3, 120.144.1-40.b.1.4, 120.144.1-40.b.1.5, 120.144.1-40.b.1.6, 120.144.1-40.b.1.7, 120.144.1-40.b.1.8, 240.144.1-40.b.1.1, 240.144.1-40.b.1.2, 240.144.1-40.b.1.3, 240.144.1-40.b.1.4, 240.144.1-40.b.1.5, 240.144.1-40.b.1.6, 240.144.1-40.b.1.7, 240.144.1-40.b.1.8, 240.144.1-40.b.1.9, 240.144.1-40.b.1.10, 240.144.1-40.b.1.11, 240.144.1-40.b.1.12, 240.144.1-40.b.1.13, 240.144.1-40.b.1.14, 240.144.1-40.b.1.15, 240.144.1-40.b.1.16, 280.144.1-40.b.1.1, 280.144.1-40.b.1.2, 280.144.1-40.b.1.3, 280.144.1-40.b.1.4, 280.144.1-40.b.1.5, 280.144.1-40.b.1.6, 280.144.1-40.b.1.7, 280.144.1-40.b.1.8
Cyclic 40-isogeny field degree: $4$
Cyclic 40-torsion field degree: $64$
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} + 3 z^{2} + 2 z w - w^{2} $
$=$ $10 x y - 10 y^{2} + 4 z^{2} - z w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 50 x^{4} - 20 x^{2} y^{2} + 40 x^{2} y z - 25 x^{2} z^{2} + 2 y^{2} z^{2} - 8 y z^{3} + 8 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{(179z^{6}-146z^{5}w-135z^{4}w^{2}+180z^{3}w^{3}-75z^{2}w^{4}+14zw^{5}-w^{6})^{3}}{z^{10}(z+w)(3z-w)^{5}(4z-w)^{2}}$

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
10.36.1.a.1 $10$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.bz.2 $40$ $3$ $3$ $1$ $0$ dimension zero
40.36.0.b.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.36.0.e.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.h.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.i.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.k.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.l.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.t.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.u.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.w.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.x.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.360.13.b.1 $40$ $5$ $5$ $13$ $0$ $1^{6}\cdot2^{3}$
80.144.3.a.2 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.b.2 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.c.1 $80$ $2$ $2$ $3$ $?$ not computed
80.144.3.d.1 $80$ $2$ $2$ $3$ $?$ not computed
80.144.7.a.2 $80$ $2$ $2$ $7$ $?$ not computed
80.144.7.b.2 $80$ $2$ $2$ $7$ $?$ not computed
80.144.7.c.2 $80$ $2$ $2$ $7$ $?$ not computed
80.144.7.d.2 $80$ $2$ $2$ $7$ $?$ not computed
120.144.5.db.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dc.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.de.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.df.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.el.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.em.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eo.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ep.1 $120$ $2$ $2$ $5$ $?$ not computed
120.216.13.b.2 $120$ $3$ $3$ $13$ $?$ not computed
120.288.13.or.2 $120$ $4$ $4$ $13$ $?$ not computed
200.360.13.b.2 $200$ $5$ $5$ $13$ $?$ not computed
240.144.3.a.2 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.b.2 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.c.1 $240$ $2$ $2$ $3$ $?$ not computed
240.144.3.d.1 $240$ $2$ $2$ $3$ $?$ not computed
240.144.7.a.2 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.b.2 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.c.2 $240$ $2$ $2$ $7$ $?$ not computed
240.144.7.d.2 $240$ $2$ $2$ $7$ $?$ not computed
280.144.5.h.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.i.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.k.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.l.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.t.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.u.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.w.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.x.1 $280$ $2$ $2$ $5$ $?$ not computed