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Properties

Label 40.192.5.ba.2
Level $40$
Index $192$
Genus $5$
Analytic rank $2$
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 $2^{6}\cdot4\cdot8$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $2$
$\Q$-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: 40.192.5.15

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}7&4\\24&11\end{bmatrix}$, $\begin{bmatrix}9&12\\12&29\end{bmatrix}$, $\begin{bmatrix}17&20\\22&3\end{bmatrix}$, $\begin{bmatrix}31&4\\18&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.384.5-40.ba.2.1, 40.384.5-40.ba.2.2, 40.384.5-40.ba.2.3, 40.384.5-40.ba.2.4, 40.384.5-40.ba.2.5, 40.384.5-40.ba.2.6, 40.384.5-40.ba.2.7, 40.384.5-40.ba.2.8, 80.384.5-40.ba.2.1, 80.384.5-40.ba.2.2, 80.384.5-40.ba.2.3, 80.384.5-40.ba.2.4, 80.384.5-40.ba.2.5, 80.384.5-40.ba.2.6, 80.384.5-40.ba.2.7, 80.384.5-40.ba.2.8, 120.384.5-40.ba.2.1, 120.384.5-40.ba.2.2, 120.384.5-40.ba.2.3, 120.384.5-40.ba.2.4, 120.384.5-40.ba.2.5, 120.384.5-40.ba.2.6, 120.384.5-40.ba.2.7, 120.384.5-40.ba.2.8, 240.384.5-40.ba.2.1, 240.384.5-40.ba.2.2, 240.384.5-40.ba.2.3, 240.384.5-40.ba.2.4, 240.384.5-40.ba.2.5, 240.384.5-40.ba.2.6, 240.384.5-40.ba.2.7, 240.384.5-40.ba.2.8, 280.384.5-40.ba.2.1, 280.384.5-40.ba.2.2, 280.384.5-40.ba.2.3, 280.384.5-40.ba.2.4, 280.384.5-40.ba.2.5, 280.384.5-40.ba.2.6, 280.384.5-40.ba.2.7, 280.384.5-40.ba.2.8
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $3840$

Jacobian

Conductor: $2^{28}\cdot5^{8}$
Simple: no
Squarefree: no
Decomposition: $1^{3}\cdot2$
Newforms: 64.2.a.a, 800.2.a.d$^{2}$, 1600.2.d.a

Models

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

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

$ 0 $ $=$ $ 16 x^{8} - 32 x^{6} z^{2} - 40 x^{4} z^{4} - 8 x^{2} z^{6} - 25 y^{4} z^{4} + z^{8} $
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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 canonical model to plane model:

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

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.v.1 :

$\displaystyle X$ $=$ $\displaystyle -2y$
$\displaystyle Y$ $=$ $\displaystyle 2x$
$\displaystyle Z$ $=$ $\displaystyle -x-t$

Equation of the image curve:

$0$ $=$ $ X^{4}+6Y^{4}-2Y^{3}Z-6Y^{2}Z^{2}-8YZ^{3}-4Z^{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
8.96.1.f.2 $8$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
40.96.1.o.2 $40$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
40.96.1.x.1 $40$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
40.96.3.u.2 $40$ $2$ $2$ $3$ $0$ $1^{2}$
40.96.3.v.1 $40$ $2$ $2$ $3$ $2$ $2$
40.96.3.z.2 $40$ $2$ $2$ $3$ $1$ $1^{2}$
40.96.3.bf.1 $40$ $2$ $2$ $3$ $1$ $1^{2}$

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.cb.2 $40$ $5$ $5$ $69$ $13$ $1^{26}\cdot2^{15}\cdot4^{2}$
40.1152.73.pp.2 $40$ $6$ $6$ $73$ $10$ $1^{28}\cdot2^{4}\cdot4^{8}$
40.1920.137.qk.1 $40$ $10$ $10$ $137$ $20$ $1^{54}\cdot2^{19}\cdot4^{10}$
80.384.13.k.2 $80$ $2$ $2$ $13$ $?$ not computed
80.384.13.n.1 $80$ $2$ $2$ $13$ $?$ not computed
80.384.13.eq.2 $80$ $2$ $2$ $13$ $?$ not computed
80.384.13.fv.2 $80$ $2$ $2$ $13$ $?$ not computed
80.384.13.hq.2 $80$ $2$ $2$ $13$ $?$ not computed
80.384.13.iv.1 $80$ $2$ $2$ $13$ $?$ not computed
80.384.13.na.1 $80$ $2$ $2$ $13$ $?$ not computed
80.384.13.nd.1 $80$ $2$ $2$ $13$ $?$ not computed
240.384.13.et.1 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.ew.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.zz.1 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.bcq.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.bjh.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.bly.1 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.chl.2 $240$ $2$ $2$ $13$ $?$ not computed
240.384.13.cho.2 $240$ $2$ $2$ $13$ $?$ not computed