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Properties

Label 40.72.1.b.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: $10$ 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: 40.72.1.56

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}13&26\\36&13\end{bmatrix}$, $\begin{bmatrix}21&38\\28&11\end{bmatrix}$, $\begin{bmatrix}29&18\\8&39\end{bmatrix}$, $\begin{bmatrix}29&20\\36&13\end{bmatrix}$, $\begin{bmatrix}37&0\\38&9\end{bmatrix}$, $\begin{bmatrix}39&30\\22&27\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.144.1-40.b.2.1, 40.144.1-40.b.2.2, 40.144.1-40.b.2.3, 40.144.1-40.b.2.4, 40.144.1-40.b.2.5, 40.144.1-40.b.2.6, 40.144.1-40.b.2.7, 40.144.1-40.b.2.8, 80.144.1-40.b.2.1, 80.144.1-40.b.2.2, 80.144.1-40.b.2.3, 80.144.1-40.b.2.4, 80.144.1-40.b.2.5, 80.144.1-40.b.2.6, 80.144.1-40.b.2.7, 80.144.1-40.b.2.8, 80.144.1-40.b.2.9, 80.144.1-40.b.2.10, 80.144.1-40.b.2.11, 80.144.1-40.b.2.12, 80.144.1-40.b.2.13, 80.144.1-40.b.2.14, 80.144.1-40.b.2.15, 80.144.1-40.b.2.16, 120.144.1-40.b.2.1, 120.144.1-40.b.2.2, 120.144.1-40.b.2.3, 120.144.1-40.b.2.4, 120.144.1-40.b.2.5, 120.144.1-40.b.2.6, 120.144.1-40.b.2.7, 120.144.1-40.b.2.8, 240.144.1-40.b.2.1, 240.144.1-40.b.2.2, 240.144.1-40.b.2.3, 240.144.1-40.b.2.4, 240.144.1-40.b.2.5, 240.144.1-40.b.2.6, 240.144.1-40.b.2.7, 240.144.1-40.b.2.8, 240.144.1-40.b.2.9, 240.144.1-40.b.2.10, 240.144.1-40.b.2.11, 240.144.1-40.b.2.12, 240.144.1-40.b.2.13, 240.144.1-40.b.2.14, 240.144.1-40.b.2.15, 240.144.1-40.b.2.16, 280.144.1-40.b.2.1, 280.144.1-40.b.2.2, 280.144.1-40.b.2.3, 280.144.1-40.b.2.4, 280.144.1-40.b.2.5, 280.144.1-40.b.2.6, 280.144.1-40.b.2.7, 280.144.1-40.b.2.8
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} + 4 z w $
$=$ $10 x y - 10 y^{2} + 2 z^{2} + 3 z w + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 5 x^{4} - 12 x^{2} z^{2} + 8 x y z^{2} - 2 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle 4y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{(269z^{6}+172z^{5}w-1760z^{4}w^{2}-3840z^{3}w^{3}-3040z^{2}w^{4}-848zw^{5}+16w^{6})^{3}}{z(z+w)^{2}(2z+w)^{10}(3z+4w)^{5}}$

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