Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x z + z t + w t + w v $ |
| $=$ | $x y - x v - y t + y v - z v + w v$ |
| $=$ | $x y + x z - x v + y u - z u$ |
| $=$ | $x w - x t - x u + x v + y w - z w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{6} z^{3} - 3 x^{5} y^{2} z^{2} - 8 x^{5} y z^{3} - 4 x^{5} z^{4} - 7 x^{4} y^{4} z + \cdots + y^{6} z^{3} $ |
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This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(1:0:0:0:0:0:0)$, $(1:0:0:1:0:1:0)$, $(0:1:1:0:0:0:0)$, $(0:1/2:1/2:-1:-2:2:1)$, $(0:0:0:1:0:0:0)$, $(0:1:1:-2:-2:-1:1)$ |
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{256x^{12}+1024x^{11}v+2048x^{10}v^{2}+3072x^{9}v^{3}+512x^{8}v^{4}+5120x^{7}v^{5}-4096x^{6}v^{6}+7168x^{5}v^{7}+1280x^{4}v^{8}-25600x^{3}v^{9}+77824x^{2}v^{10}-21316304608186326xu^{11}+262778539695610782xu^{10}v-1494553115049038654xu^{9}v^{2}+5119522572730627772xu^{8}v^{3}-11726006213200134384xu^{7}v^{4}+18800819810648402745xu^{6}v^{5}-21596771568915503789xu^{5}v^{6}+17606769055287231855xu^{4}v^{7}-10379248034163201432xu^{3}v^{8}+3209510495737411318xu^{2}v^{9}-11650752664779659631xuv^{10}+11869247426616291264xv^{11}-1132498342533576yu^{11}+38003148465587182yu^{10}v-264219503454456304yu^{9}v^{2}+954295221482456214yu^{8}v^{3}-2100709376320084950yu^{7}v^{4}+3135969377113324166yu^{6}v^{5}-3072362778117842979yu^{5}v^{6}+2393750230482852338yu^{4}v^{7}-339625126916084585yu^{3}v^{8}+2251428803776935575yu^{2}v^{9}-10643623029582758427yuv^{10}+2601495174213937104yv^{11}+65536z^{12}-65536z^{10}v^{2}-196608z^{9}v^{3}-65536z^{8}v^{4}+32768z^{7}v^{5}+217088z^{6}v^{6}+94208z^{5}v^{7}+111616z^{4}v^{8}-88064z^{3}v^{9}+71168z^{2}v^{10}-384zw^{11}-25984zw^{10}v-742272zw^{9}v^{2}-13582208zw^{8}v^{3}-191499008zw^{7}v^{4}-2310212864zw^{6}v^{5}-25437628160zw^{5}v^{6}-266020693248zw^{4}v^{7}-2705216115584zw^{3}v^{8}-27111480766848zw^{2}v^{9}-14681422123053460288zwv^{10}+9041738998594140zu^{11}-80469265497436510zu^{10}v+297714909571854788zu^{9}v^{2}-589141514097247594zu^{8}v^{3}+596981966064917506zu^{7}v^{4}-126032611497114520zu^{6}v^{5}-511913826299536633zu^{5}v^{6}+569310864107455456zu^{4}v^{7}-643774207655747733zu^{3}v^{8}-631959828669416489zu^{2}v^{9}+4170654781635489755zuv^{10}-2601495174214060496zv^{11}+16w^{12}+2624w^{11}v+107488w^{10}v^{2}+2398144w^{9}v^{3}+37967216w^{8}v^{4}+490013568w^{7}v^{5}+5605586752w^{6}v^{6}+59859249280w^{5}v^{7}+615488222960w^{4}v^{8}+6203174369344w^{3}v^{9}-8698190796316688w^{2}u^{10}+74301706063499392w^{2}u^{9}v-306080415259962528w^{2}u^{8}v^{2}+760632530181972640w^{2}u^{7}v^{3}-1283391619941422848w^{2}u^{6}v^{4}+1468285751482828288w^{2}u^{5}v^{5}-1281115129951038304w^{2}u^{4}v^{6}+512758826106399888w^{2}u^{3}v^{7}-805139058195040384w^{2}u^{2}v^{8}-1572422800865067232w^{2}uv^{9}+2440930290660763296w^{2}v^{10}+18473990462966506wu^{11}-193051764061138050wu^{10}v+975968425353262866wu^{9}v^{2}-2973290242002549108wu^{8}v^{3}+6038418512971367600wu^{7}v^{4}-8333053483098879079wu^{6}v^{5}+8153009108685982275wu^{5}v^{6}-4875161297803008017wu^{4}v^{7}+3202331504389581144wu^{3}v^{8}+2642632095629695958wu^{2}v^{9}+8030241521632378017wuv^{10}+9335438869587008336wv^{11}-10119347868925030t^{2}u^{10}+112588162349762216t^{2}u^{9}v-552479895428214102t^{2}u^{8}v^{2}+1603262059055025974t^{2}u^{7}v^{3}-3043862211044925338t^{2}u^{6}v^{4}+3983086867158717991t^{2}u^{5}v^{5}-3587518762508004030t^{2}u^{4}v^{6}+2321854130965133785t^{2}u^{3}v^{7}-722284550980047311t^{2}u^{2}v^{8}+837956071159681383t^{2}uv^{9}-3236484123973758304t^{2}v^{10}+9041738998594780tu^{11}-81258215637748538tu^{10}v+277782661890554596tu^{9}v^{2}-402865226656077862tu^{8}v^{3}-155408981737431122tu^{7}v^{4}+1623008089894653188tu^{6}v^{5}-3213864011485324891tu^{5}v^{6}+3241699495150521508tu^{4}v^{7}-2770296936640920355tu^{3}v^{8}-459204059046143623tu^{2}v^{9}-12774890114961216647tuv^{10}+3236484123973795168tv^{11}+11540504941536492u^{12}-125210943032728198u^{11}v+600630128089421290u^{10}v^{2}-1745353841476560650u^{9}v^{3}+3467747113951643584u^{8}v^{4}-5146108128011398142u^{7}v^{5}+5903846545694876877u^{6}v^{6}-5658533889558427415u^{5}v^{7}+3745522540398144069u^{4}v^{8}-3402538280975940870u^{3}v^{9}+405469757654740390u^{2}v^{10}+3294492449998964809uv^{11}+3236484123973781088v^{12}}{3823697105xu^{11}-48823032922xu^{10}v+287091033164xu^{9}v^{2}-1016464700781xu^{8}v^{3}+2404787467055xu^{7}v^{4}-3982189677748xu^{6}v^{5}+4718640092909xu^{5}v^{6}-3976634939470xu^{4}v^{7}+2393228129605xu^{3}v^{8}-835576890638xu^{2}v^{9}+1748215956941xuv^{10}-1696097135220xv^{11}+203068044yu^{11}-6906744241yu^{10}v+50305648257yu^{9}v^{2}-189001155863yu^{8}v^{3}+433488382286yu^{7}v^{4}-668991074845yu^{6}v^{5}+686841722883yu^{5}v^{6}-529921629798yu^{4}v^{7}+142967681356yu^{3}v^{8}-348436033701yu^{2}v^{9}+1516647972621yuv^{10}-369918094308yv^{11}+384zw^{6}v^{5}+26304zw^{5}v^{6}+801152zw^{4}v^{7}+16340288zw^{3}v^{8}+264308288zw^{2}v^{9}+2086767754688zwv^{10}-1621790362zu^{11}+15150031471zu^{10}v-59009704459zu^{9}v^{2}+123310140131zu^{8}v^{3}-135107714676zu^{7}v^{4}+40316826991zu^{6}v^{5}+100516896025zu^{5}v^{6}-133301215834zu^{4}v^{7}+128155686766zu^{3}v^{8}+77330257987zu^{2}v^{9}-591758355141zuv^{10}+369918094308zv^{11}-16w^{7}v^{5}-2624w^{6}v^{6}-111568w^{5}v^{7}-2715744w^{4}v^{8}-48718576w^{3}v^{9}+1560277712w^{2}u^{10}-14016166128w^{2}u^{9}v+60052872560w^{2}u^{8}v^{2}-155213941536w^{2}u^{7}v^{3}+270591892944w^{2}u^{6}v^{4}-322306090096w^{2}u^{5}v^{5}+283986562640w^{2}u^{4}v^{6}-135873314576w^{2}u^{3}v^{7}+139960787216w^{2}u^{2}v^{8}+217114231616w^{2}uv^{9}-346584981536w^{2}v^{10}-3313927551wu^{11}+36090421766wu^{10}v-188791851892wu^{9}v^{2}+596037010147wu^{8}v^{3}-1252222777505wu^{7}v^{4}+1794136482460wu^{6}v^{5}-1806405560675wu^{5}v^{6}+1156443408786wu^{4}v^{7}-665481625499wu^{3}v^{8}-323365215742wu^{2}v^{9}-1144274338979wuv^{10}-1327645026292wv^{11}+1815162489t^{2}u^{10}-20996518321t^{2}u^{9}v+107160446939t^{2}u^{8}v^{2}-322776971066t^{2}u^{7}v^{3}+635706887309t^{2}u^{6}v^{4}-860701181255t^{2}u^{5}v^{5}+804504288382t^{2}u^{4}v^{6}-530830236800t^{2}u^{3}v^{7}+192459295933t^{2}u^{2}v^{8}-142422656929t^{2}uv^{9}+462444808740t^{2}v^{10}-1621790362tu^{11}+15291586865tu^{10}v-55496570269tu^{9}v^{2}+88253834921tu^{8}v^{3}+12849722520tu^{7}v^{4}-319209265543tu^{6}v^{5}+675304397035tu^{5}v^{6}-731289160734tu^{4}v^{7}+596544589326tu^{3}v^{8}-28585902579tu^{2}v^{9}+1839985777529tuv^{10}-462444808740tv^{11}-2070047266u^{12}+23373337083u^{11}v-116676335392u^{10}v^{2}+351147061776u^{9}v^{3}-719942138111u^{8}v^{4}+1094203156373u^{7}v^{5}-1285724018880u^{6}v^{6}+1243404920453u^{5}v^{7}-878182584312u^{4}v^{8}+691589784223u^{3}v^{9}-145750542982u^{2}v^{10}-445386180747uv^{11}-462444808740v^{12}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.7.fp.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -X^{6}Z^{3}-3X^{5}Y^{2}Z^{2}-8X^{5}YZ^{3}-4X^{5}Z^{4}-7X^{4}Y^{4}Z-25X^{4}Y^{3}Z^{2}-28X^{4}Y^{2}Z^{3}-27X^{4}YZ^{4}-6X^{4}Z^{5}-4X^{3}Y^{6}+2X^{3}Y^{5}Z+20X^{3}Y^{4}Z^{2}-9X^{3}Y^{3}Z^{3}-39X^{3}Y^{2}Z^{4}-27X^{3}YZ^{5}-4X^{3}Z^{6}+12X^{2}Y^{7}+12X^{2}Y^{6}Z+12X^{2}Y^{4}Z^{3}+7X^{2}Y^{3}Z^{4}-18X^{2}Y^{2}Z^{5}-10X^{2}YZ^{6}-X^{2}Z^{7}-12XY^{8}-2XY^{7}Z+7XY^{6}Z^{2}+XY^{5}Z^{3}-XY^{4}Z^{4}+3XY^{3}Z^{5}-2XY^{2}Z^{6}-XYZ^{7}+4Y^{9}-5Y^{8}Z+Y^{7}Z^{2}+Y^{6}Z^{3} $ |
This modular curve minimally covers the modular curves listed below.
