Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x t - x u + z w + w u $ |
| $=$ | $x z + x v - z w - w t$ |
| $=$ | $ - x t + x u + y z + y v$ |
| $=$ | $x z + x t + y t - y v$ |
| $=$ | $\cdots$ |
![Copy content]()
![Toggle raw display]()
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} $ |
![Copy content]()
![Toggle raw display]()
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 |
|---|
| $(0:0:0:0:0:0:1)$, $(0:0:0:0:0:1:0)$, $(0:0:1/2:0:-1/2:-1/2:1)$, $(0:0:-1:0:1:-2:1)$ |
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{60477970049148ztu^{10}-313147307118916ztu^{9}v+6347137328149985ztu^{8}v^{2}+21501858938959120ztu^{7}v^{3}+1179725958690244ztu^{6}v^{4}-76696025313655069ztu^{5}v^{5}-62572520846522043ztu^{4}v^{6}+92840820976626584ztu^{3}v^{7}+56443435500160103ztu^{2}v^{8}-58753208416254370ztuv^{9}+18078415688777515ztv^{10}+6466168487972zu^{11}-1759171503571610zu^{10}v-7092987969660238zu^{9}v^{2}-753925356760159zu^{8}v^{3}+64982143537697127zu^{7}v^{4}+123685505028056363zu^{6}v^{5}-43730745656671857zu^{5}v^{6}-186905668349469261zu^{4}v^{7}+63239090745546938zu^{3}v^{8}+115252540086218015zu^{2}v^{9}-76833316876532632zuv^{10}+18078415653524751zv^{11}-23679778222512w^{2}u^{10}-1475390583867612w^{2}u^{9}v-1120232725541700w^{2}u^{8}v^{2}+13193316375930984w^{2}u^{7}v^{3}+31360164278132886w^{2}u^{6}v^{4}-14369945417474934w^{2}u^{5}v^{5}-65098703149767000w^{2}u^{4}v^{6}+42327802722293208w^{2}u^{3}v^{7}+61023497856678876w^{2}u^{2}v^{8}-47457383715946464w^{2}uv^{9}+13558844898539268w^{2}v^{10}-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}+147825659537100tu^{11}+478125233117546tu^{10}v-4777158739562313tu^{9}v^{2}-33400306438366646tu^{8}v^{3}-47110749190741903tu^{7}v^{4}+62790912707102524tu^{6}v^{5}+136832693487052311tu^{5}v^{6}-52824628280837008tu^{4}v^{7}-104504379943657043tu^{3}v^{8}+51972472898699032tu^{2}v^{9}-2259283228605279tuv^{10}-4519613271328262tv^{11}-4413675765625u^{12}-174307714130850u^{11}v-635509439553689u^{10}v^{2}+5190414650281317u^{9}v^{3}+34510104392479538u^{8}v^{4}+46972901375283317u^{7}v^{5}-62905573460840026u^{6}v^{6}-134049528473178766u^{5}v^{7}+51488257559135475u^{4}v^{8}+85836102164304085u^{3}v^{9}-61011968083323332u^{2}v^{10}+13558763085942617uv^{11}-282475249v^{12}}{7089267328ztu^{10}+183373312880ztu^{9}v-1355909301297ztu^{8}v^{2}+1617856038900ztu^{7}v^{3}-598115052367ztu^{6}v^{4}-357292130202ztu^{5}v^{5}+571297846714ztu^{4}v^{6}-220085997415ztu^{3}v^{7}-44736120700ztu^{2}v^{8}+32157614750ztuv^{9}-319192500ztv^{10}+2736441412zu^{11}+98334745174zu^{10}v-858827630674zu^{9}v^{2}+550790358585zu^{8}v^{3}-81138647611zu^{7}v^{4}-323776340398zu^{6}v^{5}+286406928122zu^{5}v^{6}+32649176782zu^{4}v^{7}-13349115926zu^{3}v^{8}+20389087125zu^{2}v^{9}+4090068750zuv^{10}+723714435w^{2}u^{10}+149012533596w^{2}u^{9}v-1098013594191w^{2}u^{8}v^{2}+1383976995636w^{2}u^{7}v^{3}-509193687954w^{2}u^{6}v^{4}+268159320996w^{2}u^{5}v^{5}+639423636600w^{2}u^{4}v^{6}-2300178318w^{2}u^{3}v^{7}+91323129075w^{2}u^{2}v^{8}+33284583000w^{2}uv^{9}-3042487500w^{2}v^{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}+7316561564tu^{11}-50546581435tu^{10}v+143512388435tu^{9}v^{2}+456533963607tu^{8}v^{3}-379539894607tu^{7}v^{4}+71780838108tu^{6}v^{5}+82802858831tu^{5}v^{6}-204275298230tu^{4}v^{7}-25281380138tu^{3}v^{8}-30121850525tu^{2}v^{9}-11094861000tuv^{10}+1014162500tv^{11}-7316561564u^{11}v+34087308637u^{10}v^{2}-121590258133u^{9}v^{3}-65161537772u^{8}v^{4}-73084109399u^{7}v^{5}-132630418291u^{6}v^{6}+33601391369u^{5}v^{7}+14026472008u^{4}v^{8}-1233103301u^{3}v^{9}+15979825875u^{2}v^{10}+4090068750uv^{11}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.7.jz.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
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.
