Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x^{2} - x y + y z $ |
| $=$ | $x w + x v - z w - z t + z u$ |
| $=$ | $y w + y v - z w - z u$ |
| $=$ | $x w + x u - y w - y t + y u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{6} z^{2} + 3 x^{5} y^{3} + 5 x^{5} y z^{2} - 5 x^{4} y^{2} z^{2} + 5 x^{2} y^{4} z^{2} + \cdots + y^{6} z^{2} $ |
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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 |
|---|
| $(0:0:0:0:0:0:1)$, $(0:0:0:0:1:1:0)$, $(0:0:0:1/2:-1:-1/2:1)$, $(0:0:0:-1:-1:-2:1)$ |
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{23679778222512z^{2}u^{10}+1475390583867612z^{2}u^{9}v+1120232725541700z^{2}u^{8}v^{2}-13193316375930984z^{2}u^{7}v^{3}-31360164278132886z^{2}u^{6}v^{4}+14369945417474934z^{2}u^{5}v^{5}+65098703149767000z^{2}u^{4}v^{6}-42327802722293208z^{2}u^{3}v^{7}-61023497856678876z^{2}u^{2}v^{8}+47457383715946464z^{2}uv^{9}-13558844898539268z^{2}v^{10}+60477970049148wtu^{10}-313147307118916wtu^{9}v+6347137328149985wtu^{8}v^{2}+21501858938959120wtu^{7}v^{3}+1179725958690244wtu^{6}v^{4}-76696025313655069wtu^{5}v^{5}-62572520846522043wtu^{4}v^{6}+92840820976626584wtu^{3}v^{7}+56443435500160103wtu^{2}v^{8}-58753208416254370wtuv^{9}+18078415688777515wtv^{10}-54011801561176wu^{11}-1446024196452694wu^{10}v-13440125297810223wu^{9}v^{2}-22255784295719279wu^{8}v^{3}+63802417579006883wu^{7}v^{4}+200381530341711432wu^{6}v^{5}+18841775189850186wu^{5}v^{6}-279746489326095845wu^{4}v^{7}+6795655245386835wu^{3}v^{8}+174005748502472385wu^{2}v^{9}-94911732565310147wuv^{10}+18078415653524751wv^{11}-56646589585332t^{2}u^{10}-1121171316563360t^{2}u^{9}v+3731963549688776t^{2}u^{8}v^{2}+32843298908114944t^{2}u^{7}v^{3}+47067532094890636t^{2}u^{6}v^{4}-49602874115293296t^{2}u^{5}v^{5}-115564849664321582t^{2}u^{4}v^{6}+49282471830989224t^{2}u^{3}v^{7}+79054376990764596t^{2}u^{2}v^{8}-56492373490614268t^{2}uv^{9}+13558760543665376t^{2}v^{10}+261118838707764tu^{11}+2720467866244266tu^{10}v-12241085838939865tu^{9}v^{2}-99086904254596534tu^{8}v^{3}-141245813380523175tu^{7}v^{4}+161996660937689116tu^{6}v^{5}+367962392815695475tu^{5}v^{6}-151389571942815456tu^{4}v^{7}-262613133925186235tu^{3}v^{8}+164957219879927568tu^{2}v^{9}-29376804315936031tuv^{10}-4519613271328262tv^{11}-208885924888057u^{12}-1773604263811756u^{11}v+7873612849697400u^{10}v^{2}+71434019996762907u^{9}v^{3}+128688385678112077u^{8}v^{4}-65420885447112503u^{7}v^{5}-315303116612213919u^{6}v^{6}-31942428361352534u^{5}v^{7}+235047014493557114u^{4}v^{8}-22628744225009215u^{3}v^{9}-45193924311052677u^{2}v^{10}+18078376357270879uv^{11}-282475249v^{12}}{723714435z^{2}u^{10}+149012533596z^{2}u^{9}v-1098013594191z^{2}u^{8}v^{2}+1383976995636z^{2}u^{7}v^{3}-509193687954z^{2}u^{6}v^{4}+268159320996z^{2}u^{5}v^{5}+639423636600z^{2}u^{4}v^{6}-2300178318z^{2}u^{3}v^{7}+91323129075z^{2}u^{2}v^{8}+33284583000z^{2}uv^{9}-3042487500z^{2}v^{10}-7089267328wtu^{10}-183373312880wtu^{9}v+1355909301297wtu^{8}v^{2}-1617856038900wtu^{7}v^{3}+598115052367wtu^{6}v^{4}+357292130202wtu^{5}v^{5}-571297846714wtu^{4}v^{6}+220085997415wtu^{3}v^{7}+44736120700wtu^{2}v^{8}-32157614750wtuv^{9}+319192500wtv^{10}+4352825916wu^{11}+85038567706wu^{10}v-497081670623wu^{9}v^{2}+1067065680315wu^{8}v^{3}-516976404756wu^{7}v^{4}-33515789804wu^{6}v^{5}+284890918592wu^{5}v^{6}-252735174197wu^{4}v^{7}-31387004774wu^{3}v^{8}+11768527625wu^{2}v^{9}-4409261250wuv^{10}-9142711234t^{2}u^{10}+1798891196t^{2}u^{9}v+442103941900t^{2}u^{8}v^{2}-302721224510t^{2}u^{7}v^{3}-77629897772t^{2}u^{6}v^{4}+438836422969t^{2}u^{5}v^{5}-230836877768t^{2}u^{4}v^{6}-60453393349t^{2}u^{3}v^{7}+26668401050t^{2}u^{2}v^{8}-15284855875t^{2}uv^{9}-4090068750t^{2}v^{10}+10968860904tu^{11}+46948799043tu^{10}v-1027720272235tu^{9}v^{2}+148908485413tu^{8}v^{3}+534799690151tu^{7}v^{4}-949453684046tu^{6}v^{5}+378870896705tu^{5}v^{6}+325182084928tu^{4}v^{7}-28055421962tu^{3}v^{8}+60691562275tu^{2}v^{9}+19274998500tuv^{10}-1014162500tv^{11}-1826149670u^{12}-41431128675u^{11}v+551529021698u^{10}v^{2}+275402997230u^{9}v^{3}-392008254607u^{8}v^{4}+583701370476u^{7}v^{5}-15403600646u^{6}v^{6}-298330082948u^{5}v^{7}-12639451096u^{4}v^{8}-44173603099u^{3}v^{9}-31164755625u^{2}v^{10}-3075906250uv^{11}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.7.ka.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -X^{6}Z^{2}+3X^{5}Y^{3}+5X^{5}YZ^{2}-5X^{4}Y^{2}Z^{2}+5X^{2}Y^{4}Z^{2}+5XY^{5}Z^{2}+3XY^{3}Z^{4}+Y^{6}Z^{2} $ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
