Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - x w + x u + 2 z^{2} + z w + z t - z u $ |
| $=$ | $x^{2} + x y - x u + y z - y w - z^{2} - z w$ |
| $=$ | $x z + x t + x u - y z + y t + y u + z^{2} + z w$ |
| $=$ | $x^{2} - x y + x w + y z + y t + y u - z^{2} - z w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 96 x^{8} - 32 x^{7} y - 416 x^{7} z - 4 x^{6} y^{2} + 56 x^{6} y z + 712 x^{6} z^{2} + 20 x^{5} y^{2} z + \cdots + z^{8} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 10x^{8} - 20x^{7} + 15x^{6} + 20x^{5} - 50x^{4} - 20x^{3} + 15x^{2} + 20x + 10 $ |
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This modular curve has 1 rational CM point but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2u$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2z$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x^{3}+3x^{2}z-4xz^{2}+2z^{3}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -2x^{12}+15x^{11}z-x^{11}u-39x^{10}z^{2}+14x^{10}zu-12x^{9}z^{3}-84x^{9}z^{2}u+372x^{8}z^{4}+292x^{8}z^{3}u-1236x^{7}z^{5}-652x^{7}z^{4}u+2352x^{6}z^{6}+952x^{6}z^{5}u-3000x^{5}z^{7}-840x^{5}z^{6}u+2688x^{4}z^{8}+256x^{4}z^{7}u-1728x^{3}z^{9}+352x^{3}z^{8}u+816x^{2}z^{10}-512x^{2}z^{9}u-288xz^{11}+288xz^{10}u+64z^{12}-64z^{11}u$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle x^{2}z-2xz^{2}+2z^{3}$ |
Maps to other modular curves
$j$-invariant map
of degree 80 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^{11}\,\frac{7304012579403893280000xt^{10}-14931824299190716357500xt^{9}u+90049170625432388612000xt^{8}u^{2}-33221937772868680833000xt^{7}u^{3}+48800653711280544305950xt^{6}u^{4}-144038519779164031371050xt^{5}u^{5}-62963428457311590079850xt^{4}u^{6}+90327806659849827276190xt^{3}u^{7}+70063623482753156474204xt^{2}u^{8}-43118466596242817037808xtu^{9}-37390385085055316292520xu^{10}-6030144127403881920000ywt^{9}+2943309831495395717500ywt^{8}u-78845096419869437080500ywt^{7}u^{2}-7511190204286659617000ywt^{6}u^{3}+65096141216236434037700ywt^{5}u^{4}+165682914171808376622700ywt^{4}u^{5}-45687901928841030196630ywt^{3}u^{6}-148380617031502808004630ywt^{2}u^{7}+21512026686382323601784ywtu^{8}+68903484938061011283388ywu^{9}+16883423770145219040000yt^{10}+1271577050735559070000yt^{9}u+113215205598642234278500yt^{8}u^{2}+97916070530413144055500yt^{7}u^{3}-104024111814649907125400yt^{6}u^{4}-303870829017394285875700yt^{5}u^{5}-121243430796253658250410yt^{4}u^{6}+282998965112498764111500yt^{3}u^{7}+118967680851365626953342yt^{2}u^{8}-129014544148596600496224ytu^{9}-74497528081592934030956yu^{10}+6097003967337466560000zwt^{9}-157541715012770393750zwt^{8}u+18825745381922441257750zwt^{7}u^{2}+33463689953840983661750zwt^{6}u^{3}-40552490228021338670850zwt^{5}u^{4}-113838047937435377422500zwt^{4}u^{5}+40354013774961010051500zwt^{3}u^{6}+87411471874268206030660zwt^{2}u^{7}-15315208645953238347012zwtu^{8}-42310475756858589348142zwu^{9}+8305542738741314400000zt^{10}+861457061760835446250zt^{9}u+11964154867915971441250zt^{8}u^{2}+111919850317515158421250zt^{7}u^{3}-102331128734555952807750zt^{6}u^{4}-101337902236153008107900zt^{5}u^{5}-68739116404909787525580zt^{4}u^{6}+138072840314466889855760zt^{3}u^{7}+47269476582955568870350zt^{2}u^{8}-58884993979074733556996ztu^{9}-30490172455950646161770zu^{10}+8231858306373745380000w^{2}t^{9}-9550097092417278798750w^{2}t^{8}u+65603640332957277898875w^{2}t^{7}u^{2}-8033983678180492173125w^{2}t^{6}u^{3}-33267821525546223392675w^{2}t^{5}u^{4}-117607054348228793776275w^{2}t^{4}u^{5}+36208897745659724372830w^{2}t^{3}u^{6}+100154468843104201425520w^{2}t^{2}u^{7}-17166641203657964758181w^{2}tu^{8}-46513839698194388518165w^{2}u^{9}-2027739986701917480000wt^{10}+10499990452426927406250wt^{9}u-10163590255030016348250wt^{8}u^{2}+21907910880959258741250wt^{7}u^{3}-15295856492033329610950wt^{6}u^{4}-31288843073523817372350wt^{5}u^{5}+5515543058687603441870wt^{4}u^{6}+24232545316472916815600wt^{3}u^{7}+722413275420242430006wt^{2}u^{8}-12544132125213919369054wtu^{9}-1257283286068280352802wu^{10}+6423794854738231140000t^{11}-10170730909616955167500t^{10}u+19618341975370351175375t^{9}u^{2}-5847297318615677259375t^{8}u^{3}-40407848821280288400150t^{7}u^{4}-57451448750695170860150t^{6}u^{5}+33246652951839241504565t^{5}u^{6}+132137527862221251852425t^{4}u^{7}-29537793773042856339733t^{3}u^{8}-92208479447875358014237t^{2}u^{9}+10552677359011576450097tu^{10}+33436253111218164978991u^{11}}{18264188379729702115625xt^{10}+36320548326311350184375xt^{9}u+174173869131609286690000xt^{8}u^{2}+38676805741364727713250xt^{7}u^{3}-184036186234836264984700xt^{6}u^{4}-402665295035739330116400xt^{5}u^{5}+36994846161107432193600xt^{4}u^{6}+291271577356544397304280xt^{3}u^{7}+95054055243484802283648xt^{2}u^{8}-152341462929116589168656xtu^{9}-82465164268410146127360xu^{10}-27231758461105700612500ywt^{9}-101467940847844684780625ywt^{8}u-187445764340175355473750ywt^{7}u^{2}+65549693402579183009250ywt^{6}u^{3}+452035206417667864157000ywt^{5}u^{4}+229447693832471955334400ywt^{4}u^{5}-312046854463493720031960ywt^{3}u^{6}-289894216532172399006760ywt^{2}u^{7}+151465744283305757865568ywtu^{8}+152508239946131300401296ywu^{9}+53876854940259071750000yt^{10}+190210902818370469529375yt^{9}u+381554159890795328803125yt^{8}u^{2}+120075576197063922410000yt^{7}u^{3}-796704872380924863109950yt^{6}u^{4}-855203596427505745191400yt^{5}u^{5}+276926697791000902461080yt^{4}u^{6}+763973983136623499522800yt^{3}u^{7}+71478506424212922045624yt^{2}u^{8}-396941637871252902389808ytu^{9}-164993754276563459697328yu^{10}+18476159293298443003125zwt^{9}+69711808499362746806250zwt^{8}u+95429388630874072773750zwt^{7}u^{2}+262987567363987321250zwt^{6}u^{3}-345802502733507767137400zwt^{5}u^{4}-97564925561146102645000zwt^{4}u^{5}+183994357102546831250600zwt^{3}u^{6}+180155556988280276968920zwt^{2}u^{7}-98103787035988723747344zwtu^{8}-93312363940105098639504zwu^{9}+21439730101837243928125zt^{10}+88320989215613599794375zt^{9}u+144180630043816406598750zt^{8}u^{2}+110499531551754480332750zt^{7}u^{3}-426348024863153888790550zt^{6}u^{4}-323146048969722558850200zt^{5}u^{5}+97051695306721694914840zt^{4}u^{6}+357164435593668066657320zt^{3}u^{7}+18201710636869904726360zt^{2}u^{8}-176682206485091721294192ztu^{9}-67412581779923427671024zu^{10}+24222426246166183359375w^{2}t^{9}+61933914720397072061875w^{2}t^{8}u+133203125274730072888750w^{2}t^{7}u^{2}-43221527762563132466500w^{2}t^{6}u^{3}-330727440748706147084400w^{2}t^{5}u^{4}-142983799179289777472800w^{2}t^{4}u^{5}+218263860612397037826960w^{2}t^{3}u^{6}+194085805308531997402040w^{2}t^{2}u^{7}-107066771114472425115032w^{2}tu^{8}-102767949380075724830840w^{2}u^{9}-100800640793388137500wt^{10}+29552193813490227560000wt^{9}u+22893049865428578985000wt^{8}u^{2}-3285110698776199813750wt^{7}u^{3}-74685080515487053691600wt^{6}u^{4}-54659428232027657512100wt^{5}u^{5}+49081325776335965718440wt^{4}u^{6}+56727366877390218441800wt^{3}u^{7}-18737103570382656034368wt^{2}u^{8}-29994681163594188500128wtu^{9}-2800428354973663078928wu^{10}+11453426830101354446875t^{11}+17396299899919669785625t^{10}u+42205703175820580556875t^{9}u^{2}-39275216778466630463125t^{8}u^{3}-247137961196617533288950t^{7}u^{4}-38876003498523327430000t^{6}u^{5}+323403788074393818295380t^{5}u^{6}+187681517742915776927800t^{4}u^{7}-189416769365059924523976t^{3}u^{8}-187009184779202496510064t^{2}u^{9}+73387737660848868928680tu^{10}+73998250111179313935304u^{11}}$ 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Cover information
Click on a modular curve in the diagram to see information about it.
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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.
