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Properties

Label 40.120.8.da.1
Level $40$
Index $120$
Genus $8$
Analytic rank $2$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $40$ $\SL_2$-level: $40$ Newform level: $800$
Index: $120$ $\PSL_2$-index:$120$
Genus: $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $5^{2}\cdot10\cdot20\cdot40^{2}$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $2$
$\Q$-gonality: $3 \le \gamma \le 5$
$\overline{\Q}$-gonality: $3 \le \gamma \le 5$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 40C8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.120.8.31

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}5&32\\29&35\end{bmatrix}$, $\begin{bmatrix}19&24\\39&1\end{bmatrix}$, $\begin{bmatrix}23&20\\31&19\end{bmatrix}$, $\begin{bmatrix}27&24\\26&23\end{bmatrix}$, $\begin{bmatrix}39&0\\28&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.240.8-40.da.1.1, 40.240.8-40.da.1.2, 40.240.8-40.da.1.3, 40.240.8-40.da.1.4, 40.240.8-40.da.1.5, 40.240.8-40.da.1.6, 40.240.8-40.da.1.7, 40.240.8-40.da.1.8, 40.240.8-40.da.1.9, 40.240.8-40.da.1.10, 40.240.8-40.da.1.11, 40.240.8-40.da.1.12, 40.240.8-40.da.1.13, 40.240.8-40.da.1.14, 40.240.8-40.da.1.15, 40.240.8-40.da.1.16, 80.240.8-40.da.1.1, 80.240.8-40.da.1.2, 80.240.8-40.da.1.3, 80.240.8-40.da.1.4, 80.240.8-40.da.1.5, 80.240.8-40.da.1.6, 80.240.8-40.da.1.7, 80.240.8-40.da.1.8, 80.240.8-40.da.1.9, 80.240.8-40.da.1.10, 80.240.8-40.da.1.11, 80.240.8-40.da.1.12, 80.240.8-40.da.1.13, 80.240.8-40.da.1.14, 80.240.8-40.da.1.15, 80.240.8-40.da.1.16, 120.240.8-40.da.1.1, 120.240.8-40.da.1.2, 120.240.8-40.da.1.3, 120.240.8-40.da.1.4, 120.240.8-40.da.1.5, 120.240.8-40.da.1.6, 120.240.8-40.da.1.7, 120.240.8-40.da.1.8, 120.240.8-40.da.1.9, 120.240.8-40.da.1.10, 120.240.8-40.da.1.11, 120.240.8-40.da.1.12, 120.240.8-40.da.1.13, 120.240.8-40.da.1.14, 120.240.8-40.da.1.15, 120.240.8-40.da.1.16, 240.240.8-40.da.1.1, 240.240.8-40.da.1.2, 240.240.8-40.da.1.3, 240.240.8-40.da.1.4, 240.240.8-40.da.1.5, 240.240.8-40.da.1.6, 240.240.8-40.da.1.7, 240.240.8-40.da.1.8, 240.240.8-40.da.1.9, 240.240.8-40.da.1.10, 240.240.8-40.da.1.11, 240.240.8-40.da.1.12, 240.240.8-40.da.1.13, 240.240.8-40.da.1.14, 240.240.8-40.da.1.15, 240.240.8-40.da.1.16, 280.240.8-40.da.1.1, 280.240.8-40.da.1.2, 280.240.8-40.da.1.3, 280.240.8-40.da.1.4, 280.240.8-40.da.1.5, 280.240.8-40.da.1.6, 280.240.8-40.da.1.7, 280.240.8-40.da.1.8, 280.240.8-40.da.1.9, 280.240.8-40.da.1.10, 280.240.8-40.da.1.11, 280.240.8-40.da.1.12, 280.240.8-40.da.1.13, 280.240.8-40.da.1.14, 280.240.8-40.da.1.15, 280.240.8-40.da.1.16
Cyclic 40-isogeny field degree: $6$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $6144$

Jacobian

Conductor: $2^{26}\cdot5^{16}$
Simple: no
Squarefree: no
Decomposition: $1^{4}\cdot2^{2}$
Newforms: 50.2.a.b$^{3}$, 200.2.a.e, 800.2.d.a, 800.2.d.c

Models

Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations

$ 0 $ $=$ $ x r - y u - 2 z u - w r $
$=$ $2 x t + y t - z u - w u - w r$
$=$ $x^{2} + x w + y z + 2 w^{2}$
$=$ $x u - y u + z t - z u + z v - w t - w v$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 34848 x^{14} + 956 x^{12} y^{2} + 3904 x^{12} y z - 3655 x^{12} z^{2} - 252 x^{10} y^{4} + 5366 x^{10} y^{3} z + \cdots - 32 z^{14} $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:0:1/2:0:-1/2:1)$, $(0:0:0:0:1:-2:1:0)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle w$
$\displaystyle Y$ $=$ $\displaystyle v$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}r$

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 40.60.4.bl.1 :

$\displaystyle X$ $=$ $\displaystyle -x$
$\displaystyle Y$ $=$ $\displaystyle -y$
$\displaystyle Z$ $=$ $\displaystyle w$
$\displaystyle W$ $=$ $\displaystyle -z$

Equation of the image curve:

$0$ $=$ $ X^{2}-XZ+2Z^{2}+YW $
$=$ $ 2X^{2}Z-Y^{2}Z-2Z^{3}+XYW-YZW+2XW^{2}+2ZW^{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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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_{S_4}(5)$ $5$ $24$ $24$ $0$ $0$ full Jacobian
8.24.0.ba.2 $8$ $5$ $5$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.ba.2 $8$ $5$ $5$ $0$ $0$ full Jacobian
40.60.4.bl.1 $40$ $2$ $2$ $4$ $0$ $2^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.240.16.bk.1 $40$ $2$ $2$ $16$ $5$ $1^{4}\cdot2^{2}$
40.240.16.br.2 $40$ $2$ $2$ $16$ $4$ $1^{4}\cdot2^{2}$
40.240.16.bt.2 $40$ $2$ $2$ $16$ $3$ $1^{4}\cdot2^{2}$
40.240.16.bu.1 $40$ $2$ $2$ $16$ $6$ $1^{4}\cdot2^{2}$
40.240.16.by.1 $40$ $2$ $2$ $16$ $3$ $1^{4}\cdot2^{2}$
40.240.16.ca.2 $40$ $2$ $2$ $16$ $4$ $1^{4}\cdot2^{2}$
40.240.16.cc.2 $40$ $2$ $2$ $16$ $3$ $1^{4}\cdot2^{2}$
40.240.16.ce.2 $40$ $2$ $2$ $16$ $4$ $1^{4}\cdot2^{2}$
40.360.22.gq.1 $40$ $3$ $3$ $22$ $2$ $1^{6}\cdot4^{2}$
40.480.29.baa.1 $40$ $4$ $4$ $29$ $5$ $1^{9}\cdot2^{2}\cdot4^{2}$
80.240.16.cg.2 $80$ $2$ $2$ $16$ $?$ not computed
80.240.16.ci.1 $80$ $2$ $2$ $16$ $?$ not computed
80.240.16.co.1 $80$ $2$ $2$ $16$ $?$ not computed
80.240.16.cq.1 $80$ $2$ $2$ $16$ $?$ not computed
80.240.16.cu.2 $80$ $2$ $2$ $16$ $?$ not computed
80.240.16.da.1 $80$ $2$ $2$ $16$ $?$ not computed
80.240.16.dc.1 $80$ $2$ $2$ $16$ $?$ not computed
80.240.16.di.1 $80$ $2$ $2$ $16$ $?$ not computed
80.240.17.bw.1 $80$ $2$ $2$ $17$ $?$ not computed
80.240.17.cc.1 $80$ $2$ $2$ $17$ $?$ not computed
80.240.17.ce.1 $80$ $2$ $2$ $17$ $?$ not computed
80.240.17.ck.2 $80$ $2$ $2$ $17$ $?$ not computed
80.240.17.cu.1 $80$ $2$ $2$ $17$ $?$ not computed
80.240.17.cw.1 $80$ $2$ $2$ $17$ $?$ not computed
80.240.17.dc.1 $80$ $2$ $2$ $17$ $?$ not computed
80.240.17.de.2 $80$ $2$ $2$ $17$ $?$ not computed
120.240.16.eo.1 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.es.1 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.ew.1 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fa.1 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.fy.2 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.gc.2 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.gg.1 $120$ $2$ $2$ $16$ $?$ not computed
120.240.16.gk.2 $120$ $2$ $2$ $16$ $?$ not computed
240.240.16.dq.2 $240$ $2$ $2$ $16$ $?$ not computed
240.240.16.ds.2 $240$ $2$ $2$ $16$ $?$ not computed
240.240.16.eg.2 $240$ $2$ $2$ $16$ $?$ not computed
240.240.16.ei.2 $240$ $2$ $2$ $16$ $?$ not computed
240.240.16.fi.2 $240$ $2$ $2$ $16$ $?$ not computed
240.240.16.fs.2 $240$ $2$ $2$ $16$ $?$ not computed
240.240.16.fy.2 $240$ $2$ $2$ $16$ $?$ not computed
240.240.16.gi.2 $240$ $2$ $2$ $16$ $?$ not computed
240.240.17.em.2 $240$ $2$ $2$ $17$ $?$ not computed
240.240.17.ew.2 $240$ $2$ $2$ $17$ $?$ not computed
240.240.17.fc.2 $240$ $2$ $2$ $17$ $?$ not computed
240.240.17.fm.1 $240$ $2$ $2$ $17$ $?$ not computed
240.240.17.im.1 $240$ $2$ $2$ $17$ $?$ not computed
240.240.17.io.1 $240$ $2$ $2$ $17$ $?$ not computed
240.240.17.jc.2 $240$ $2$ $2$ $17$ $?$ not computed
240.240.17.je.1 $240$ $2$ $2$ $17$ $?$ not computed
280.240.16.eu.1 $280$ $2$ $2$ $16$ $?$ not computed
280.240.16.ey.2 $280$ $2$ $2$ $16$ $?$ not computed
280.240.16.fc.2 $280$ $2$ $2$ $16$ $?$ not computed
280.240.16.fg.1 $280$ $2$ $2$ $16$ $?$ not computed
280.240.16.fk.2 $280$ $2$ $2$ $16$ $?$ not computed
280.240.16.fo.2 $280$ $2$ $2$ $16$ $?$ not computed
280.240.16.fs.1 $280$ $2$ $2$ $16$ $?$ not computed
280.240.16.fw.2 $280$ $2$ $2$ $16$ $?$ not computed