Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ z v + t u $ |
| $=$ | $x v + y u$ |
| $=$ | $x v + y v + w u$ |
| $=$ | $ - x t + y z$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 4 x^{4} y^{3} - 6 x^{3} y^{4} - 8 x^{3} y^{2} z^{2} - 2 x^{3} z^{4} + x^{2} y^{5} - x^{2} y z^{4} + \cdots + y^{5} z^{2} $ |
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This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:-1:0:1:0:0)$, $(0:0:1:0:0:0:0)$, $(2:-1:0:-1/2:0:2:1)$, $(0:0:0:-1/2:1:0:0)$, $(0:0:0:1/2:1:0:0)$, $(-2:1:0:1/2:0:2:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{5}\cdot\frac{100000z^{12}+1200000z^{11}u+5200000z^{11}v+4200000z^{10}u^{2}+27100000z^{10}uv+67000000z^{10}v^{2}+19600000z^{9}u^{2}v+90400000z^{9}uv^{2}-31600000z^{9}v^{3}+86000000z^{8}u^{2}v^{2}+347700000z^{8}uv^{3}+4149000000z^{8}v^{4}-856800000z^{7}u^{2}v^{3}-3952000000z^{7}uv^{4}-92747600000z^{7}v^{5}+20577900000z^{6}u^{2}v^{4}+108165400000z^{6}uv^{5}+2486666800000z^{6}v^{6}-537511600000z^{5}u^{2}v^{5}-3059814400000z^{5}uv^{6}-71781774800000z^{5}v^{7}+15279074500000z^{4}u^{2}v^{6}+91851946074992z^{4}uv^{7}+2187373541165288z^{4}v^{8}-460423088449472z^{3}u^{2}v^{7}-2877789379260064z^{3}uv^{8}-68347364974616800z^{3}v^{9}+14429615842603628z^{2}u^{2}v^{8}+82488319121215970z^{2}uv^{9}+1233555744405269191z^{2}v^{10}+511974992zt^{11}+6155649600zt^{10}v-15849251976zt^{9}v^{2}-1749807318656zt^{8}v^{3}-34733263997884zt^{7}v^{4}-459841331742032zt^{6}v^{5}-4734908436394066zt^{5}v^{6}-39225973302433808zt^{4}v^{7}-249258189375938259zt^{3}v^{8}-888128094059650300zt^{2}v^{9}+1021464287896206073ztv^{10}-355555373594262368zu^{2}v^{9}+533333060401353552zuv^{10}+1244443807526998288zv^{11}-2047499968w^{2}t^{10}-24656998400w^{2}t^{9}v+67834407840w^{2}t^{8}v^{2}+7052563071360w^{2}t^{7}v^{3}+138814539873520w^{2}t^{6}v^{4}+1825017376050560w^{2}t^{5}v^{5}+18662502281888520w^{2}t^{4}v^{6}+153483833593402400w^{2}t^{3}v^{7}+969179349588915980w^{2}t^{2}v^{8}+3555553735850623680w^{2}tv^{9}+24000000w^{2}v^{10}+511974992t^{12}+6163249600t^{11}v-15431376984t^{10}v^{2}-1745529969024t^{9}v^{3}-34769896823412t^{8}v^{4}-461550637909616t^{7}v^{5}-4765958423520646t^{6}v^{6}-39652607242034096t^{5}v^{7}-254973128652555201t^{4}v^{8}-986634882109130004t^{3}v^{9}-468815909753255741t^{2}v^{10}+888888433952655920tv^{11}+2800000u^{2}v^{10}-11600000uv^{11}+6100000v^{12}}{v^{2}(80000z^{5}v^{5}-20000z^{4}u^{2}v^{4}-40000z^{4}uv^{5}-1099992z^{4}v^{6}+240000z^{3}u^{2}v^{5}+1359504z^{3}uv^{6}+31676376z^{3}v^{7}-6784596z^{2}u^{2}v^{6}-37319942z^{2}uv^{7}-560700789z^{2}v^{8}+8zt^{9}+400zt^{8}v+9796zt^{7}v^{2}+159656zt^{6}v^{3}+1985974zt^{5}v^{4}+17400772zt^{4}v^{5}+112953049zt^{3}v^{6}+403012614zt^{2}v^{7}-464400883ztv^{8}+161802596zu^{2}v^{7}-242703894zuv^{8}-566309086zv^{9}-32w^{2}t^{8}-1600w^{2}t^{7}v-39120w^{2}t^{6}v^{2}-635360w^{2}t^{5}v^{3}-7863320w^{2}t^{4}v^{4}-68614960w^{2}t^{3}v^{5}-441041380w^{2}t^{2}v^{6}-1618025960w^{2}tv^{7}+8t^{10}+400t^{9}v+9804t^{8}v^{2}+160024t^{7}v^{3}+1994322t^{6}v^{4}+17608076t^{5}v^{5}+115612291t^{4}v^{6}+448423442t^{3}v^{7}+213344847t^{2}v^{8}-404506490tv^{9})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.7.ga.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -4X^{4}Y^{3}-6X^{3}Y^{4}-8X^{3}Y^{2}Z^{2}-2X^{3}Z^{4}+X^{2}Y^{5}-X^{2}YZ^{4}+4XY^{6}+4XY^{4}Z^{2}+Y^{7}+Y^{5}Z^{2} $ |
This modular curve minimally covers the modular curves listed below.
