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Properties

Label 40.24.1.cf.2
Level $40$
Index $24$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $10$ Newform level: $80$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (none of which are rational) Cusp widths $2^{2}\cdot10^{2}$ Cusp orbits $2^{2}$
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: 10D1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.24.1.150

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&15\\4&7\end{bmatrix}$, $\begin{bmatrix}10&31\\7&34\end{bmatrix}$, $\begin{bmatrix}24&19\\13&35\end{bmatrix}$, $\begin{bmatrix}32&37\\21&18\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.48.1-40.cf.2.1, 40.48.1-40.cf.2.2, 40.48.1-40.cf.2.3, 40.48.1-40.cf.2.4, 80.48.1-40.cf.2.1, 80.48.1-40.cf.2.2, 80.48.1-40.cf.2.3, 80.48.1-40.cf.2.4, 80.48.1-40.cf.2.5, 80.48.1-40.cf.2.6, 80.48.1-40.cf.2.7, 80.48.1-40.cf.2.8, 120.48.1-40.cf.2.1, 120.48.1-40.cf.2.2, 120.48.1-40.cf.2.3, 120.48.1-40.cf.2.4, 240.48.1-40.cf.2.1, 240.48.1-40.cf.2.2, 240.48.1-40.cf.2.3, 240.48.1-40.cf.2.4, 240.48.1-40.cf.2.5, 240.48.1-40.cf.2.6, 240.48.1-40.cf.2.7, 240.48.1-40.cf.2.8, 280.48.1-40.cf.2.1, 280.48.1-40.cf.2.2, 280.48.1-40.cf.2.3, 280.48.1-40.cf.2.4
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $30720$

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 $ $=$ $ 10 x^{2} - y w $
$=$ $125 y^{2} - 22 y w - 10 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 25 x^{4} - 44 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{10}w$

Maps to other modular curves

$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^3\,\frac{133593750yz^{4}w+1957095000yz^{2}w^{3}+14770296yw^{5}-1953125z^{6}-257737500z^{4}w^{2}-268262820z^{2}w^{4}-1111968w^{6}}{w(1562500yz^{4}+1502500yz^{2}w^{2}+68381yw^{4}+550000z^{4}w-3520z^{2}w^{3}-5148w^{5})}$

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
20.12.1.a.1 $20$ $2$ $2$ $1$ $0$ dimension zero
40.12.0.bp.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.12.0.bx.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
40.72.1.j.1 $40$ $3$ $3$ $1$ $0$ dimension zero
40.96.5.j.2 $40$ $4$ $4$ $5$ $0$ $1^{2}\cdot2$
40.120.5.cf.1 $40$ $5$ $5$ $5$ $0$ $1^{2}\cdot2$
80.48.3.dt.1 $80$ $2$ $2$ $3$ $?$ not computed
80.48.3.du.1 $80$ $2$ $2$ $3$ $?$ not computed
80.48.3.dx.1 $80$ $2$ $2$ $3$ $?$ not computed
80.48.3.dy.1 $80$ $2$ $2$ $3$ $?$ not computed
120.72.5.iz.1 $120$ $3$ $3$ $5$ $?$ not computed
120.96.5.er.2 $120$ $4$ $4$ $5$ $?$ not computed
200.120.5.j.2 $200$ $5$ $5$ $5$ $?$ not computed
240.48.3.fp.1 $240$ $2$ $2$ $3$ $?$ not computed
240.48.3.fq.1 $240$ $2$ $2$ $3$ $?$ not computed
240.48.3.ft.2 $240$ $2$ $2$ $3$ $?$ not computed
240.48.3.fu.2 $240$ $2$ $2$ $3$ $?$ not computed
280.192.13.ez.2 $280$ $8$ $8$ $13$ $?$ not computed