Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x w - 2 x t + y w - y t - 2 z w - z t $ |
| $=$ | $10 y^{2} + 2 z^{2} - 3 w^{2} - 2 w t - 2 t^{2}$ |
| $=$ | $18 x^{2} - 28 x y + 4 x z + 2 y^{2} - 12 y z - 2 z^{2} - 5 t^{2}$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{8} - 64 x^{7} z - 328 x^{6} y^{2} + 72 x^{6} z^{2} + 784 x^{5} y^{2} z + 8 x^{5} z^{3} + 681 x^{4} y^{4} + \cdots + z^{8} $ |
![Copy content]()
![Toggle raw display]()
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Maps to other modular curves
$j$-invariant map
of degree 80 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 3^3\,\frac{8092112702767430106861780363929053440xyz^{9}+105694053591435403431600909976689043200xyz^{7}t^{2}+138877929064649918341955270239068896000xyz^{5}t^{4}-677787101616569583955950530397130380000xyz^{3}t^{6}-2004076834745677820401124930323506000000xyzt^{8}+1120899895508188071018231567657551424xz^{10}-23419511140566630612732257893582440000xz^{8}t^{2}-219832191197826935419106245099785736400xz^{6}t^{4}-478984695407418873382140047767816344000xz^{4}t^{6}-120692178857150760842070326882095762500xz^{2}t^{8}+2456149205535718405425378664677685417674xt^{10}+5826581640063626631938767009804598208yz^{10}+78871418956759805229017986356209472000yz^{8}t^{2}+241173944382129814723062751084594597200yz^{6}t^{4}+209522464590739512111290820377698822000yz^{4}t^{6}-118816617813795897660109643672202737500yz^{2}t^{8}+596559134029734831486726334620976230144yw^{10}+790229655032358266295679531431810959616yw^{9}t+2708915445117092484559237333954220998272yw^{8}t^{2}+810506095844954830521018054445442482304yw^{7}t^{3}+2694490125012743865011818697382305571968yw^{6}t^{4}+1194882795166270442033092605835567059552yw^{5}t^{5}-868084803229640418397412667361074209704yw^{4}t^{6}-967807081948010484331359043436619616328yw^{3}t^{7}-533306774259817831376204086119965249711yw^{2}t^{8}-574424530169798860568583995110875860727ywt^{9}+1316005275982588779298896122048247562276yt^{10}+2819356026300643862675108939531668416z^{11}+8935641128149590061830004891618645440z^{9}t^{2}-41101346404494576758668784010511786800z^{7}t^{4}-180205867867552208541034389894304112000z^{5}t^{6}-190939967245550843233149975590037157500z^{3}t^{8}-979714342533174512877062051832243876096zw^{10}-3795762739134975701962460432287563636480zw^{9}t-9134849375389408910501211458790084301440zw^{8}t^{2}-14767728718166606441576077350038804148160zw^{7}t^{3}-19969336852727258185654773033199588298480zw^{6}t^{4}-23253186760944464037733733074045488575184zw^{5}t^{5}-18951625169401968757043533605441036588280zw^{4}t^{6}-12841842893096598626569027705900974640120zw^{3}t^{7}-4192267947345740808549833565037743835400zw^{2}t^{8}+1372002235545839898589705446696454473905zwt^{9}+803389712032990222446137407797738061648zt^{10}}{126439260980741095419715318186391460xyz^{9}-941302501014050535225791982974303700xyz^{7}t^{2}-487190826343250539207336317126623500xyz^{5}t^{4}+2595254147410652438936385470676442500xyz^{3}t^{6}-2052982756339531633922166010442250000xyzt^{8}+17514060867315438609659868244649241xz^{10}+386554417669737267024140432904107250xz^{8}t^{2}-1263418287349410915262322525639657600xz^{6}t^{4}-624365987314034398458420876871102875xz^{4}t^{6}+104626492208843027977895200494496875xz^{2}t^{8}+120072449068227778554076056575245406118xt^{10}+91040338125994166124043234528196847yz^{10}-738473328411658202640003064877199375yz^{8}t^{2}-290596560625564919679280642596066450yz^{6}t^{4}+1758788931470213052903512264687390500yz^{4}t^{6}-99165690544995835824383675954246875yz^{2}t^{8}-9480128983782702568105094991414117yw^{10}-132836844365695755569391611303420793yw^{9}t-327218360026046387096936314390853511yw^{8}t^{2}-1206678095502488628266533425552709267yw^{7}t^{3}-1691892231542626175928398072421604309yw^{6}t^{4}-1208629608701012763316241564246264241yw^{5}t^{5}-4604917155777773657308087831893997888yw^{4}t^{6}-7213841848640387902296243843637298276yw^{3}t^{7}-14518697367265427106373359797918107517yw^{2}t^{8}-30136770035881150820071861050529325109ywt^{9}+60046885072764266305541730918772902247yt^{10}+44052437910947560354298577180182319z^{11}+140665708439073394277941570080815460z^{9}t^{2}-744237979694729945231547199714174950z^{7}t^{4}-330874186595208036934345198980233000z^{5}t^{6}+78208798694387559710354876657004375z^{3}t^{8}+3237014640337240912182979037468730zw^{10}+149787655568341840964045650704064695zw^{9}t+334536341353492536898726541632934130zw^{8}t^{2}+2043223599073116431135646678958307465zw^{7}t^{3}+4857403953623327798134415700853972000zw^{6}t^{4}+8315340336778136963818257344496990195zw^{5}t^{5}+20097961497454404517213533374113188500zw^{4}t^{6}+37651990619451824732406121344509266880zw^{3}t^{7}+76573659725740718125058229781882236670zw^{2}t^{8}+150372863490783253248653923036531971005zwt^{9}+59760075598649372785388730947158514644zt^{10}}$ |
Hi
|
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.
