Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ - x t + y w + y u $ |
| $=$ | $x w + x t + x u + z t$ |
| $=$ | $x^{2} + x y + y z$ |
| $=$ | $x w - x s - y w + y u - y s$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{9} y - 2 x^{8} y^{2} + 5 x^{8} z^{2} + 2 x^{7} y^{3} + 15 x^{7} y z^{2} + 4 x^{6} y^{4} + \cdots + 5 y^{8} z^{2} $ |
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This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:-1/2:0:1/2:1/5:9/10:1)$, $(0:0:0:0:0:0:0:1/2:1)$, $(0:0:0:0:1:0:0:1:0)$, $(0:0:0:0:0:0:1/2:1:0)$, $(0:0:0:1:0:-1:-1:0:1)$, $(0:0:0:1:3:2:-1:0:1)$, $(0:0:0:0:0:0:-2:1:0)$, $(0:0:0:1:-3:2:1/5:-18/5:1)$ |
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{3^3}{2^6}\cdot\frac{1647044200360112640wr^{9}+2147556489896037440wr^{8}s-1035869621414038608wr^{7}s^{2}+996348505729934986wr^{6}s^{3}+40538612076868520wr^{5}s^{4}-1627454323264298556wr^{4}s^{5}+362645198379935424wr^{3}s^{6}-525611025909403740wr^{2}s^{7}-82905171545782566wrs^{8}-6052824037340172ws^{9}+2400067205149955712tvr^{8}+2189887297285834944tvr^{7}s-4178687965416323688tvr^{6}s^{2}+128747808246151386tvr^{5}s^{3}+1083294991450075521tvr^{4}s^{4}-2161324019538545337tvr^{3}s^{5}+1021030660528348005tvr^{2}s^{6}+224299479268032777tvrs^{7}-5750471083487715tvs^{8}+635000441182287104tr^{9}-522387946033929152tr^{8}s-3027291681555494336tr^{7}s^{2}+462416640822606514tr^{6}s^{3}+759041491691754473tr^{5}s^{4}-743291756290697400tr^{4}s^{5}+1068930283763089692tr^{3}s^{6}+291237226650699138tr^{2}s^{7}+13527541104312150trs^{8}+7933701897063423ts^{9}+1103452388685416064uvr^{8}+1640745728991566368uvr^{7}s-823048412152168976uvr^{6}s^{2}-223285610512760643uvr^{5}s^{3}+194281662017595396uvr^{4}s^{4}-871712621921243124uvr^{3}s^{5}+1387116659710854uvr^{2}s^{6}-25833999818972268uvrs^{7}-21346648508781585uvs^{8}-1345242488850579712ur^{9}+2101271722650411136ur^{8}s+5992587931093653488ur^{7}s^{2}-4822922462344133860ur^{6}s^{3}-692652377277296681ur^{5}s^{4}+2418035146315559052ur^{4}s^{5}-3445663039838382852ur^{3}s^{6}+832044865432782690ur^{2}s^{7}+177211737289241424urs^{8}+23320669056249357us^{9}+635935963875359744v^{2}r^{8}+1614752288553500096v^{2}r^{7}s+532247280190707312v^{2}r^{6}s^{2}-660219338505822874v^{2}r^{5}s^{3}+14043189091256229v^{2}r^{4}s^{4}-376364880529108173v^{2}r^{3}s^{5}-528029105879026431v^{2}r^{2}s^{6}-8407222464530475v^{2}rs^{7}+19502584277964813v^{2}s^{8}+952090418408543616vr^{9}-4663649401064208608vr^{8}s-7446584329689670032vr^{7}s^{2}+8396160763361383043vr^{6}s^{3}+434380497986826280vr^{5}s^{4}-2970533288042775711vr^{4}s^{5}+5216707677468943269vr^{3}s^{6}-1697340622021597935vr^{2}s^{7}-336516081745195746vrs^{8}-33191649429477921vs^{9}-634946744430527744r^{10}+1867557972732669120r^{9}s+3207168320614206528r^{8}s^{2}-4194367773787532578r^{7}s^{3}+420603767568593195r^{6}s^{4}+1513997279403705181r^{5}s^{5}-2582204662576348773r^{4}s^{6}+1132148657960478855r^{3}s^{7}-5827961326708413r^{2}s^{8}-26818498548817884rs^{9}-1937177352338436s^{10}}{12337769241216wr^{9}+11610883894688wr^{8}s-11806729941472wr^{7}s^{2}+14480336552910wr^{6}s^{3}-12599339242408wr^{5}s^{4}-3636369300wr^{4}s^{5}-1145400936192wr^{3}s^{6}-2029102321572wr^{2}s^{7}-26986302306wrs^{8}-1692845892ws^{9}+17993640418752tvr^{8}+9825207502752tvr^{7}s-34628385355224tvr^{6}s^{2}+15079003415478tvr^{5}s^{3}-5536544442525tvr^{4}s^{4}-3855979156491tvr^{3}s^{5}+4671657496575tvr^{2}s^{6}+64027105371tvrs^{7}+464597775tvs^{8}+4765058455552tr^{9}-5657307165984tr^{8}s-20552385438816tr^{7}s^{2}+11138550718478tr^{6}s^{3}-2533309100589tr^{5}s^{4}+1729011813336tr^{4}s^{5}+4857225204708tr^{3}s^{6}+161965404630tr^{2}s^{7}+11230159626trs^{8}-3767516955ts^{9}+8260572804352uvr^{8}+9300001562864uvr^{7}s-9478927714760uvr^{6}s^{2}+4791812664655uvr^{5}s^{3}-7762935801300uvr^{4}s^{4}+1799753181300uvr^{3}s^{5}-946641593358uvr^{2}s^{6}-30549460716uvrs^{7}-6024644811uvs^{8}-10086609471616ur^{9}+19429778026624ur^{8}s+37693754095008ur^{7}s^{2}-51037050869652ur^{6}s^{3}+24273688114637ur^{5}s^{4}-13288038584700ur^{4}s^{5}-3758553864540ur^{3}s^{6}+3517920632358ur^{2}s^{7}+24725934936urs^{8}-1792530081us^{9}+4765058455552v^{2}r^{8}+10405521711648v^{2}r^{7}s+182034654608v^{2}r^{6}s^{2}-7902441198758v^{2}r^{5}s^{3}+10068224516559v^{2}r^{4}s^{4}-11729559406599v^{2}r^{3}s^{5}+534533276115v^{2}r^{2}s^{6}+22505572575v^{2}rs^{7}+5460362847v^{2}s^{8}+7147587683328vr^{9}-37573718358128vr^{8}s-42015212504408vr^{7}s^{2}+79170232515489vr^{6}s^{3}-39750719327024vr^{5}s^{4}+25550789852787vr^{4}s^{5}+2744892339471vr^{3}s^{6}-6745833866709vr^{2}s^{7}-57255475806vrs^{8}-1028879739vs^{9}-4765058455552r^{10}+15743916637600r^{9}s+18225435108960r^{8}s^{2}-38113669629230r^{7}s^{3}+22570155649121r^{6}s^{4}-11163954531049r^{5}s^{5}-1556055725055r^{4}s^{6}+3597610120533r^{3}s^{7}-616553918991r^{2}s^{8}-2113290324rs^{9}-564281964s^{10}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.9.fs.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -X^{9}Y-2X^{8}Y^{2}+5X^{8}Z^{2}+2X^{7}Y^{3}+15X^{7}YZ^{2}+4X^{6}Y^{4}-15X^{6}Y^{2}Z^{2}-2X^{5}Y^{5}-30X^{5}Y^{3}Z^{2}-2X^{4}Y^{6}+40X^{4}Y^{4}Z^{2}+X^{3}Y^{7}+30X^{3}Y^{5}Z^{2}-225X^{3}Y^{3}Z^{4}-15X^{2}Y^{6}Z^{2}-15XY^{7}Z^{2}+225XY^{5}Z^{4}+5Y^{8}Z^{2} $ |
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.
