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Properties

Label 40.192.5.a.1
Level $40$
Index $192$
Genus $5$
Analytic rank $0$
Cusps $24$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $1600$
Index: $192$ $\PSL_2$-index:$192$
Genus: $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps: $24$ (none of which are rational) Cusp widths $8^{24}$ Cusp orbits $4^{4}\cdot8$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8A5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.192.5.40

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}5&4\\8&1\end{bmatrix}$, $\begin{bmatrix}19&8\\36&35\end{bmatrix}$, $\begin{bmatrix}27&26\\36&1\end{bmatrix}$, $\begin{bmatrix}37&38\\20&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.384.5-40.a.1.1, 40.384.5-40.a.1.2, 40.384.5-40.a.1.3, 40.384.5-40.a.1.4, 40.384.5-40.a.1.5, 40.384.5-40.a.1.6, 40.384.5-40.a.1.7, 40.384.5-40.a.1.8, 80.384.5-40.a.1.1, 80.384.5-40.a.1.2, 80.384.5-40.a.1.3, 80.384.5-40.a.1.4, 120.384.5-40.a.1.1, 120.384.5-40.a.1.2, 120.384.5-40.a.1.3, 120.384.5-40.a.1.4, 120.384.5-40.a.1.5, 120.384.5-40.a.1.6, 120.384.5-40.a.1.7, 120.384.5-40.a.1.8, 240.384.5-40.a.1.1, 240.384.5-40.a.1.2, 240.384.5-40.a.1.3, 240.384.5-40.a.1.4, 280.384.5-40.a.1.1, 280.384.5-40.a.1.2, 280.384.5-40.a.1.3, 280.384.5-40.a.1.4, 280.384.5-40.a.1.5, 280.384.5-40.a.1.6, 280.384.5-40.a.1.7, 280.384.5-40.a.1.8
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $3840$

Jacobian

Conductor: $2^{30}\cdot5^{4}$
Simple: no
Squarefree: no
Decomposition: $1^{3}\cdot2$
Newforms: 64.2.a.a, 64.2.b.a, 1600.2.a.n$^{2}$

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $ $=$ $ 2 x^{2} + y^{2} + z^{2} $
$=$ $5 y^{2} - 5 z^{2} - t^{2}$
$=$ $10 y z - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 625 x^{8} - 1000 x^{6} z^{2} + 50 x^{4} y^{4} + 1400 x^{4} z^{4} - 40 x^{2} y^{4} z^{2} - 800 x^{2} z^{6} + \cdots + 144 z^{8} $
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Rational points

This modular curve has no real points and no $\Q_p$ points for $p=29$, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x+y$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}t$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 40.96.3.t.1 :

$\displaystyle X$ $=$ $\displaystyle -x$
$\displaystyle Y$ $=$ $\displaystyle 2x+w$
$\displaystyle Z$ $=$ $\displaystyle 2x+t$

Equation of the image curve:

$0$ $=$ $ 68X^{4}-32X^{3}Y-24X^{2}Y^{2}-8XY^{3}-Y^{4}-32X^{3}Z-24X^{2}Z^{2}-8XZ^{3}-Z^{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
8.96.3.c.1 $8$ $2$ $2$ $3$ $0$ $1^{2}$
40.96.1.a.1 $40$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
40.96.1.a.2 $40$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
40.96.1.l.1 $40$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
40.96.3.r.1 $40$ $2$ $2$ $3$ $0$ $1^{2}$
40.96.3.r.2 $40$ $2$ $2$ $3$ $0$ $1^{2}$
40.96.3.t.1 $40$ $2$ $2$ $3$ $0$ $2$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.960.69.y.1 $40$ $5$ $5$ $69$ $17$ $1^{26}\cdot2^{15}\cdot4^{2}$
40.1152.73.kx.1 $40$ $6$ $6$ $73$ $10$ $1^{28}\cdot2^{4}\cdot4^{8}$
40.1920.137.hh.1 $40$ $10$ $10$ $137$ $26$ $1^{54}\cdot2^{19}\cdot4^{10}$
80.384.17.er.1 $80$ $2$ $2$ $17$ $?$ not computed
80.384.17.er.2 $80$ $2$ $2$ $17$ $?$ not computed
80.384.17.es.1 $80$ $2$ $2$ $17$ $?$ not computed
80.384.17.es.2 $80$ $2$ $2$ $17$ $?$ not computed
240.384.17.bgl.1 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.bgl.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.bgm.1 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.bgm.2 $240$ $2$ $2$ $17$ $?$ not computed