Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w - x v - y w $ |
| $=$ | $y z + y w + t u$ |
| $=$ | $y w - y v - w u$ |
| $=$ | $x z + x w + y t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{8} + 8 x^{7} y - 10 x^{6} y^{2} - x^{6} z^{2} + 8 x^{5} y z^{2} + 10 x^{4} y^{4} + \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 |
|---|
| $(0:0:1:0:0:0:1)$, $(0:0:-1:1:-1:0:1)$, $(0:0:0:-1:1:0:0)$, $(0:0:0:1/2:1/2:0:1)$ |
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{28858971668086784xtu^{9}v+75333504757579776xtu^{7}v^{3}+59853989896379392xtu^{5}v^{5}+4894303539966118080xtu^{3}v^{7}-4529705399547229236xtuv^{9}-14732133006311424xu^{11}-171902732342263808xu^{9}v^{2}-10449490633973760xu^{7}v^{4}-2355609434846600704xu^{5}v^{6}+12621277485924287264xu^{3}v^{8}-23995506162206915202xuv^{10}-67426669035520000ytu^{9}v-28627329942159360ytu^{7}v^{3}+10429133369499648ytu^{5}v^{5}+344225134669688320ytu^{3}v^{7}-19421130156181557872ytuv^{9}+13463554623537152yu^{11}+112042032045490176yu^{9}v^{2}+25092643744563200yu^{7}v^{4}+1928958249890208768yu^{5}v^{6}-7848528904093253184yu^{3}v^{8}+8819557435774465364yuv^{10}+10257932288524288ztu^{10}+66847895368171520ztu^{8}v^{2}-18673067770302464ztu^{6}v^{4}+1290199209345322752ztu^{4}v^{6}-7839622738142988048ztu^{2}v^{8}+14625479072183769773ztv^{10}+28688387685482496zu^{10}v-9972416372948992zu^{8}v^{3}-7376547083219968zu^{6}v^{5}+389220469636585408zu^{4}v^{7}+7762699266546417452zu^{2}v^{9}+5230409045292814567zv^{11}+37931166655523654443wtv^{10}+19559243305975808wu^{10}v-104700334677016576wu^{8}v^{3}-24562372571630592wu^{6}v^{5}-967270489704169408wu^{4}v^{7}+14573533559631413556wu^{2}v^{9}+10460818152025629134wv^{11}+4096000000t^{12}+40960000000t^{11}v+151552000000t^{10}v^{2}+262144000000t^{9}v^{3}+245760000000t^{8}v^{4}+84197468176384t^{7}v^{5}-8179452553089024t^{6}v^{6}-95476296101742080t^{5}v^{7}-622993688145340736t^{4}v^{8}-4919071144379142840t^{3}v^{9}+2686913413644288t^{2}u^{10}+98333621794439168t^{2}u^{8}v^{2}+70594973623373824t^{2}u^{6}v^{4}+1018835532521611520t^{2}u^{4}v^{6}-3024731307403998960t^{2}u^{2}v^{8}+847118316290797935t^{2}v^{10}-40543234785017856tu^{10}v-41500986268483584tu^{8}v^{3}-11950674056472576tu^{6}v^{5}+122217227196966784tu^{4}v^{7}-13591866861635987176tu^{2}v^{9}-18075279000895070103tv^{11}+1268578386968576u^{12}+12171067680751616u^{10}v^{2}+55799169798234112u^{8}v^{4}+489406619300836864u^{6}v^{6}-1382584960707359904u^{4}v^{8}-11268208370011581854u^{2}v^{10}-5230409041196814567v^{12}}{13205504000xtu^{9}v+414829264896xtu^{7}v^{3}-297735486464xtu^{5}v^{5}-7008541784768xtu^{3}v^{7}+6470660037652xtuv^{9}+80740352000xu^{11}+451912269824xu^{9}v^{2}-264625561600xu^{7}v^{4}+2187274530304xu^{5}v^{6}-23947059050016xu^{3}v^{8}+48516790050354xuv^{10}+296366899200ytu^{9}v+323113975808ytu^{7}v^{3}-253963395072ytu^{5}v^{5}-3589763923456ytu^{3}v^{7}+39121382236400ytuv^{9}-71565312000yu^{11}-331545575424yu^{9}v^{2}+454399967232yu^{7}v^{4}-1764975787008yu^{5}v^{6}+14942792467520yu^{3}v^{8}-19461895884852yuv^{10}-53346304000ztu^{10}-324908285952ztu^{8}v^{2}+92951236608ztu^{6}v^{4}-934630127360ztu^{4}v^{6}+14258036411536ztu^{2}v^{8}-29232010032037ztv^{10}-88493260800zu^{10}v+34688942080zu^{8}v^{3}-37596376064zu^{6}v^{5}+349092613696zu^{4}v^{7}-15876012863564zu^{2}v^{9}-10794739665103zv^{11}-77340949178483wtv^{10}-39341785088wu^{10}v+316235890688wu^{8}v^{3}-204832527360wu^{6}v^{5}+1556401371584wu^{4}v^{7}-28969296988436wu^{2}v^{9}-21589479330206wv^{11}-478208000t^{7}v^{5}+7083876352t^{6}v^{6}+143554340352t^{5}v^{7}+1008535476032t^{4}v^{8}+9406218094072t^{3}v^{9}-19398656000t^{2}u^{10}+34528755712t^{2}u^{8}v^{2}+221567889408t^{2}u^{6}v^{4}-1454575745280t^{2}u^{4}v^{6}+6342968512880t^{2}u^{2}v^{8}-2261064791063t^{2}v^{10}+179070566400tu^{10}v+153105367040tu^{8}v^{3}-224010524672tu^{6}v^{5}-1926754220928tu^{4}v^{7}+26813034416808tu^{2}v^{9}+37314199481343tv^{11}-9175040000u^{12}+134944260096u^{10}v^{2}+58576216064u^{8}v^{4}-668251424256u^{6}v^{6}+1749028653984u^{4}v^{8}+22629657804014u^{2}v^{10}+10794739665103v^{12}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.7.fn.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -2X^{8}+8X^{7}Y-10X^{6}Y^{2}-X^{6}Z^{2}+8X^{5}YZ^{2}+10X^{4}Y^{4}-5X^{4}Y^{2}Z^{2}-8X^{3}Y^{5}+2X^{3}YZ^{4}+2X^{2}Y^{6}+5X^{2}Y^{4}Z^{2}+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.
