Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
$ 0 $ | $=$ | $ x z + y^{2} $ |
| $=$ | $x w - x u + y v$ |
| $=$ | $y w - y u - z v$ |
| $=$ | $x w - x t + x u + y t + y u$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 10 x^{7} y^{2} + x^{7} z^{2} - 50 x^{6} y^{3} - 5 x^{6} y z^{2} + 90 x^{5} y^{4} + 9 x^{5} y^{2} z^{2} + \cdots + 5 y^{7} z^{2} $ |
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Canonical model |
$(0:0:0:1:0:0:0)$, $(0:0:0:1:2:1:1)$, $(0:0:0:0:0:-1:1)$, $(0:0:0:-2/3:-2/3:1/3:1)$ |
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 -2^2\cdot3\,\frac{8766986554081280z^{2}v^{10}+1934917632w^{12}-7739670528w^{11}v-23219011584w^{10}v^{2}+61917364224w^{9}v^{3}+371504185344w^{8}v^{4}+309586821120w^{7}v^{5}-3374496350208w^{6}v^{6}-15355506327552w^{5}v^{7}-13064563851264w^{4}v^{8}+152440550719488w^{3}v^{9}+781892475420672w^{2}v^{10}-120272597523wu^{11}-363106413711wu^{10}v+17740078020171wu^{9}v^{2}-72635338352817wu^{8}v^{3}-73922997043854wu^{7}v^{4}-174564082775430wu^{6}v^{5}+38995441267878wu^{5}v^{6}+1083415763387214wu^{4}v^{7}+2239684764676497wu^{3}v^{8}+1934138910616965wu^{2}v^{9}+226174741379103wuv^{10}-636987672331789wv^{11}+7558272t^{12}+30233088t^{11}v-60466176t^{10}v^{2}-60466176t^{9}v^{3}-181398528t^{8}v^{4}-3869835264t^{7}v^{5}+24912064512t^{6}v^{6}-130969737216t^{5}v^{7}+773180992512t^{4}v^{8}-4627355516928t^{3}v^{9}+24568458381t^{2}u^{10}-1119969116658t^{2}u^{9}v+10223363381049t^{2}u^{8}v^{2}-18506194143192t^{2}u^{7}v^{3}+7083000912186t^{2}u^{6}v^{4}-102679332413580t^{2}u^{5}v^{5}-146806300089414t^{2}u^{4}v^{6}-64039005833112t^{2}u^{3}v^{7}+292322587833657t^{2}u^{2}v^{8}+636103265754606t^{2}uv^{9}-145028637817331t^{2}v^{10}+15919551351tu^{11}+447090389235tu^{10}v-13318243256331tu^{9}v^{2}+57894245845713tu^{8}v^{3}-11249172378378tu^{7}v^{4}+21925437777486tu^{6}v^{5}-229002873198534tu^{5}v^{6}-937796938619598tu^{4}v^{7}-1231187601083517tu^{3}v^{8}-242543775589713tu^{2}v^{9}-170710199372831tuv^{10}+636863837603341tv^{11}+22302216810u^{12}+1868415664050u^{11}v-16894097004906u^{10}v^{2}+11940027683742u^{9}v^{3}+90906139716228u^{8}v^{4}+348092318671380u^{7}v^{5}+800194321793484u^{6}v^{6}+815477211574812u^{5}v^{7}-539257098794382u^{4}v^{8}-3159385167551334u^{3}v^{9}-1880004157910978u^{2}v^{10}+437003627738086uv^{11}+30958682112v^{12}}{230784696320z^{2}v^{10}+23219011584w^{2}v^{10}+8017850367wu^{11}-49928881950wu^{10}v-81723402657wu^{9}v^{2}+59177018232wu^{8}v^{3}+173243034414wu^{7}v^{4}+145405800540wu^{6}v^{5}+116891865126wu^{5}v^{6}+94110406632wu^{4}v^{7}-23687919693wu^{3}v^{8}-451825823790wu^{2}v^{9}+300814552731wuv^{10}-58866551920wv^{11}+90699264t^{7}v^{5}-544195584t^{6}v^{6}+2902376448t^{5}v^{7}-17051461632t^{4}v^{8}+101673874944t^{3}v^{9}+3360065247t^{2}u^{10}-1525609647t^{2}u^{9}v-28221117984t^{2}u^{8}v^{2}-18397131480t^{2}u^{7}v^{3}+17443644678t^{2}u^{6}v^{4}+13819703058t^{2}u^{5}v^{5}-39912481080t^{2}u^{4}v^{6}-142795783008t^{2}u^{3}v^{7}-232717032189t^{2}u^{2}v^{8}-204661617195t^{2}uv^{9}+35647540336t^{2}v^{10}-4621588083tu^{11}+31784777622tu^{10}v+21854802537tu^{9}v^{2}-108958159584tu^{8}v^{3}-157070221902tu^{7}v^{4}-44663010228tu^{6}v^{5}+112357610514tu^{5}v^{6}+314630078112tu^{4}v^{7}+558518096097tu^{3}v^{8}+536867721198tu^{2}v^{9}-521073222299tuv^{10}+58866551920tv^{11}-7981653330u^{12}+2609099748u^{11}v+99527942430u^{10}v^{2}+187509635856u^{9}v^{3}+100733660316u^{8}v^{4}-82120125672u^{7}v^{5}-210521406996u^{6}v^{6}-254676181968u^{5}v^{7}-170588339082u^{4}v^{8}+81189176484u^{3}v^{9}+153720195382u^{2}v^{10}+30378794272uv^{11}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.7.fq.2
:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
$0$ |
$=$ |
$ 10X^{7}Y^{2}+X^{7}Z^{2}-50X^{6}Y^{3}-5X^{6}YZ^{2}+90X^{5}Y^{4}+9X^{5}Y^{2}Z^{2}-50X^{4}Y^{5}+15X^{4}Y^{3}Z^{2}+2X^{4}YZ^{4}-50X^{3}Y^{6}-29X^{3}Y^{4}Z^{2}-2X^{3}Y^{2}Z^{4}+90X^{2}Y^{7}+25X^{2}Y^{5}Z^{2}-50XY^{8}-5XY^{6}Z^{2}+10Y^{9}+5Y^{7}Z^{2} $ |
This modular curve minimally covers the modular curves listed below.