Properties

Label 80.240.8-40.db.1.9
Level $80$
Index $240$
Genus $8$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $80$ $\SL_2$-level: $80$ Newform level: $800$
Index: $240$ $\PSL_2$-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: not computed
$\Q$-gonality: $3 \le \gamma \le 8$
$\overline{\Q}$-gonality: $3 \le \gamma \le 8$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 40C8

Level structure

$\GL_2(\Z/80\Z)$-generators: $\begin{bmatrix}9&4\\48&77\end{bmatrix}$, $\begin{bmatrix}10&13\\13&74\end{bmatrix}$, $\begin{bmatrix}54&67\\67&30\end{bmatrix}$, $\begin{bmatrix}67&74\\12&73\end{bmatrix}$, $\begin{bmatrix}75&54\\64&57\end{bmatrix}$
Contains $-I$: no $\quad$ (see 40.120.8.db.1 for the level structure with $-I$)
Cyclic 80-isogeny field degree: $12$
Cyclic 80-torsion field degree: $192$
Full 80-torsion field degree: $49152$

Models

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

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

$ 0 $ $=$ $ 40000000 x^{14} + 812500 x^{12} y^{2} + 13750000 x^{12} y z + 28250000 x^{12} z^{2} + 12500 x^{10} y^{4} + \cdots + 8 y z^{13} $
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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
$(0:0:0:0:-1/2:1/2:1/2:1)$, $(0:0:0:0:1:1:1:0)$

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} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.120.8.db.1 :

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

Equation of the image curve:

$0$ $=$ $ 40000000X^{14}+812500X^{12}Y^{2}+12500X^{10}Y^{4}+13750000X^{12}YZ+187500X^{10}Y^{3}Z+28250000X^{12}Z^{2}+1762500X^{10}Y^{2}Z^{2}-17500X^{8}Y^{4}Z^{2}+11275000X^{10}YZ^{3}-175000X^{8}Y^{3}Z^{3}-1500X^{6}Y^{5}Z^{3}+5100000X^{10}Z^{4}+1480625X^{8}Y^{2}Z^{4}-17625X^{6}Y^{4}Z^{4}+450X^{4}Y^{6}Z^{4}+1327500X^{8}YZ^{5}+169125X^{6}Y^{3}Z^{5}+4800X^{4}Y^{5}Z^{5}+60X^{2}Y^{7}Z^{5}+652500X^{8}Z^{6}+315500X^{6}Y^{2}Z^{6}+30375X^{4}Y^{4}Z^{6}+655X^{2}Y^{6}Z^{6}+2Y^{8}Z^{6}+317250X^{6}YZ^{7}+64700X^{4}Y^{3}Z^{7}+2905X^{2}Y^{5}Z^{7}+22Y^{7}Z^{7}+114500X^{6}Z^{8}+71225X^{4}Y^{2}Z^{8}+6155X^{2}Y^{4}Z^{8}+94Y^{6}Z^{8}+35550X^{4}YZ^{9}+6975X^{2}Y^{3}Z^{9}+202Y^{5}Z^{9}+5600X^{4}Z^{10}+4090X^{2}Y^{2}Z^{10}+240Y^{4}Z^{10}+1080X^{2}YZ^{11}+160Y^{3}Z^{11}+80X^{2}Z^{12}+56Y^{2}Z^{12}+8YZ^{13} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
80.48.0-40.ca.2.12 $80$ $5$ $5$ $0$ $?$
80.120.4-40.bl.1.5 $80$ $2$ $2$ $4$ $?$
80.120.4-40.bl.1.14 $80$ $2$ $2$ $4$ $?$

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

Covering curve Level Index Degree Genus
80.480.16-40.bl.2.3 $80$ $2$ $2$ $16$
80.480.16-40.bq.2.8 $80$ $2$ $2$ $16$
80.480.16-40.bs.2.4 $80$ $2$ $2$ $16$
80.480.16-40.bv.1.4 $80$ $2$ $2$ $16$
80.480.16-40.by.1.4 $80$ $2$ $2$ $16$
80.480.16-40.cb.2.8 $80$ $2$ $2$ $16$
80.480.16-40.cd.2.4 $80$ $2$ $2$ $16$
80.480.16-40.ce.1.4 $80$ $2$ $2$ $16$
80.480.16-80.ce.2.3 $80$ $2$ $2$ $16$
80.480.16-80.ck.1.9 $80$ $2$ $2$ $16$
80.480.16-80.cm.1.9 $80$ $2$ $2$ $16$
80.480.16-80.cs.1.3 $80$ $2$ $2$ $16$
80.480.16-80.cw.2.5 $80$ $2$ $2$ $16$
80.480.16-80.cy.1.9 $80$ $2$ $2$ $16$
80.480.16-80.de.1.9 $80$ $2$ $2$ $16$
80.480.16-80.dg.1.5 $80$ $2$ $2$ $16$
80.480.17-80.by.1.5 $80$ $2$ $2$ $17$
80.480.17-80.ca.1.9 $80$ $2$ $2$ $17$
80.480.17-80.cg.1.9 $80$ $2$ $2$ $17$
80.480.17-80.ci.2.5 $80$ $2$ $2$ $17$
80.480.17-80.cs.1.3 $80$ $2$ $2$ $17$
80.480.17-80.cy.1.9 $80$ $2$ $2$ $17$
80.480.17-80.da.1.9 $80$ $2$ $2$ $17$
80.480.17-80.dg.2.3 $80$ $2$ $2$ $17$
240.480.16-240.dm.1.3 $240$ $2$ $2$ $16$
240.480.16-240.dw.1.3 $240$ $2$ $2$ $16$
240.480.16-240.ec.2.3 $240$ $2$ $2$ $16$
240.480.16-240.em.2.5 $240$ $2$ $2$ $16$
240.480.16-120.ep.1.16 $240$ $2$ $2$ $16$
240.480.16-120.et.2.9 $240$ $2$ $2$ $16$
240.480.16-120.ex.1.16 $240$ $2$ $2$ $16$
240.480.16-120.fb.2.11 $240$ $2$ $2$ $16$
240.480.16-240.fm.2.4 $240$ $2$ $2$ $16$
240.480.16-240.fo.2.4 $240$ $2$ $2$ $16$
240.480.16-120.fz.2.13 $240$ $2$ $2$ $16$
240.480.16-240.gc.2.4 $240$ $2$ $2$ $16$
240.480.16-120.ge.2.11 $240$ $2$ $2$ $16$
240.480.16-240.ge.2.6 $240$ $2$ $2$ $16$
240.480.16-120.gi.2.13 $240$ $2$ $2$ $16$
240.480.16-120.gl.1.9 $240$ $2$ $2$ $16$
240.480.17-240.eq.2.4 $240$ $2$ $2$ $17$
240.480.17-240.es.2.4 $240$ $2$ $2$ $17$
240.480.17-240.fg.2.4 $240$ $2$ $2$ $17$
240.480.17-240.fi.1.6 $240$ $2$ $2$ $17$
240.480.17-240.ii.2.4 $240$ $2$ $2$ $17$
240.480.17-240.is.2.4 $240$ $2$ $2$ $17$
240.480.17-240.iy.2.4 $240$ $2$ $2$ $17$
240.480.17-240.ji.1.6 $240$ $2$ $2$ $17$