Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} - 3 x w - x t - 3 y z + 6 y w + 6 y t + z w $ |
| $=$ | $4 x^{2} + 15 x y - x w + 15 y z + w^{2}$ |
| $=$ | $5 x^{2} - 15 x y + 8 x z - 5 x w - 2 x t + 27 y z + 6 y w + 6 y t - 4 z^{2} + z w + z t + 2 w^{2} - t^{2}$ |
| $=$ | $3 x^{2} + 4 x z + 2 x w + 90 y^{2} - 6 y z - 3 y w - 3 y t - z w + w t$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2806 x^{6} - 2909 x^{5} y - 4849 x^{5} z + 3191 x^{4} y^{2} + 14367 x^{4} y z + 16001 x^{4} z^{2} + \cdots + 36 z^{6} $ |
![Copy content]()
![Toggle raw display]()
Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ -27x^{6} + 81x^{5} - 105x^{4} + 75x^{3} - 45x^{2} + 21x - 22 $ |
![Copy content]()
![Toggle raw display]()
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{30178211}{1080613600}y^{7}-\frac{9722117}{1620920400}y^{6}w-\frac{120023}{170623200}y^{6}t+\frac{11125607}{9725522400}y^{5}w^{2}+\frac{250241}{972552240}y^{5}wt+\frac{13293}{1080613600}y^{5}t^{2}-\frac{1593719}{14588283600}y^{4}w^{3}-\frac{633}{27015340}y^{4}w^{2}t-\frac{3829}{405230100}y^{4}wt^{2}-\frac{131}{202615050}y^{4}t^{3}+\frac{119}{475704900}y^{3}w^{4}+\frac{15791}{5470606350}y^{3}w^{3}t+\frac{4831}{5470606350}y^{3}w^{2}t^{2}+\frac{226}{2735303175}y^{3}wt^{3}+\frac{11519}{16411819050}y^{2}w^{5}+\frac{7}{26175150}y^{2}w^{4}t+\frac{118}{2735303175}y^{2}w^{3}t^{2}+\frac{16}{8205909525}y^{2}w^{2}t^{3}-\frac{59}{1491983550}yw^{6}-\frac{26}{2735303175}yw^{5}t-\frac{8}{8205909525}yw^{4}t^{2}+\frac{2}{2735303175}w^{7}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{262058551487}{2926820323328000}y^{21}+\frac{57389598217351481}{515667692306190336000}y^{20}w+\frac{29970280319}{462129524736000}y^{20}t+\frac{292736127645597767601}{5047441316908375373824000}y^{19}w^{2}+\frac{238299595772661}{9046801619406848000}y^{19}wt-\frac{11267771851}{26341382909952000}y^{19}t^{2}+\frac{1924147609520721045883}{454269718521753783644160000}y^{18}w^{3}+\frac{337300173683816371}{103951880668593543168000}y^{18}w^{2}t+\frac{64856738432297}{185640369230228520960}y^{18}wt^{2}+\frac{187407397}{4939009295616000}y^{18}t^{3}+\frac{681828234177387411787}{1362809155565261350932480000}y^{17}w^{4}+\frac{18823338447472640173}{23908932553776514928640000}y^{17}w^{3}t+\frac{154069572658279740631}{817685493339156810559488000}y^{17}w^{2}t^{2}+\frac{4201032855551}{290063076922232064000}y^{17}wt^{3}+\frac{486201312766335940811}{2044213733347892026398720000}y^{16}w^{5}+\frac{100388083451935341581}{817685493339156810559488000}y^{16}w^{4}t+\frac{62679675502218469457}{12265282400087352158392320000}y^{16}w^{3}t^{2}+\frac{76158418379819779}{153316030001091901979904000}y^{16}w^{2}t^{3}+\frac{13416122580971209307}{4181346272757051872179200000}y^{15}w^{6}+\frac{1191540116432472109}{278756418183803458145280000}y^{15}w^{5}t+\frac{2685282148954490189}{12265282400087352158392320000}y^{15}w^{4}t^{2}+\frac{169083798795210181}{2299740450016378529698560000}y^{15}w^{3}t^{3}+\frac{274908877038282007657}{110387541600786169425530880000}y^{14}w^{7}+\frac{485550877675419055439}{137984427000982711781913600000}y^{14}w^{6}t+\frac{25711972074647422187}{27596885400196542356382720000}y^{14}w^{5}t^{2}+\frac{462899110512119107}{6899221350049135589095680000}y^{14}w^{4}t^{3}+\frac{19238915060545919527}{137984427000982711781913600000}y^{13}w^{8}+\frac{266198829519578057}{7262338263209616409574400000}y^{13}w^{7}t-\frac{24716967329037479021}{1655813124011792541382963200000}y^{13}w^{6}t^{2}-\frac{4662547585247}{5988907421917652421090000}y^{13}w^{5}t^{3}-\frac{2700972815571676271}{261444177475546190744678400000}y^{12}w^{9}+\frac{8151842224212919}{12936040031342129229554400000}y^{12}w^{8}t+\frac{124343837986011463}{310464960752211101509305600000}y^{12}w^{7}t^{2}+\frac{4155447399611323}{34496106750245677945478400000}y^{12}w^{6}t^{3}+\frac{3644058937713695957}{931394882256633304527916800000}y^{11}w^{10}+\frac{2165536681927434493}{465697441128316652263958400000}y^{11}w^{9}t+\frac{225750765233289407}{186278976451326660905583360000}y^{11}w^{8}t^{2}+\frac{18978394751694049}{232848720564158326131979200000}y^{11}w^{7}t^{3}-\frac{134482157001132989}{931394882256633304527916800000}y^{10}w^{11}-\frac{138926993691716473}{931394882256633304527916800000}y^{10}w^{10}t-\frac{4977117663333697}{127008393034995450617443200000}y^{10}w^{9}t^{2}-\frac{44012664168841}{17463654042311874459898440000}y^{10}w^{8}t^{3}-\frac{34908704171796229}{1397092323384949956791875200000}y^{9}w^{12}-\frac{2378707192793081}{232848720564158326131979200000}y^{9}w^{11}t-\frac{102379328586499}{69854616169247497839593760000}y^{9}w^{10}t^{2}-\frac{3122120916971}{87318270211559372299492200000}y^{9}w^{9}t^{3}+\frac{38554592316391}{13970923233849499567918752000}y^{8}w^{13}+\frac{1890277266953}{759289306187472802604280000}y^{8}w^{12}t+\frac{13117738108951}{21829567552889843074873050000}y^{8}w^{11}t^{2}+\frac{21238718173}{574462304023416923022975000}y^{8}w^{10}t^{3}-\frac{2617109163601}{26195481063467811689847660000}y^{7}w^{14}-\frac{9324250689191}{87318270211559372299492200000}y^{7}w^{13}t-\frac{4351760393503}{174636540423118744598984400000}y^{7}w^{12}t^{2}-\frac{14310800909}{8186087832333691153077393750}y^{7}w^{11}t^{3}-\frac{2822770088197}{174636540423118744598984400000}y^{6}w^{15}-\frac{154863460627}{16372175664667382306154787500}y^{6}w^{14}t-\frac{127365873163}{65488702658669529224619150000}y^{6}w^{13}t^{2}-\frac{327219221}{2976759211757705873846325000}y^{6}w^{12}t^{3}+\frac{34017293213}{39293221595201717534771490000}y^{5}w^{16}+\frac{26825073947}{49116526994002146918464362500}y^{5}w^{15}t+\frac{2635356311}{24558263497001073459232181250}y^{5}w^{14}t^{2}+\frac{2151371}{327443513293347646123095750}y^{5}w^{13}t^{3}+\frac{975783289}{49116526994002146918464362500}y^{4}w^{17}+\frac{520525853}{98233053988004293836928725000}y^{4}w^{16}t+\frac{13290413}{14734958098200644075539308750}y^{4}w^{15}t^{2}+\frac{1547342}{36837395245501610188848271875}y^{4}w^{14}t^{3}-\frac{3848543}{2232569408818279405384743750}y^{3}w^{18}-\frac{16931417}{24558263497001073459232181250}y^{3}w^{17}t-\frac{842228}{7367479049100322037769654375}y^{3}w^{16}t^{2}-\frac{10504}{1601625880239200442993403125}y^{3}w^{15}t^{3}+\frac{292288}{12279131748500536729616090625}y^{2}w^{19}+\frac{44794}{4093043916166845576538696875}y^{2}w^{18}t+\frac{8}{5243757330320513905885875}y^{2}w^{17}t^{2}+\frac{128}{1473495809820064407553930875}y^{2}w^{16}t^{3}+\frac{1336}{12279131748500536729616090625}yw^{20}-\frac{136}{4093043916166845576538696875}yw^{19}t-\frac{32}{36837395245501610188848271875}yw^{18}t^{2}-\frac{32}{12279131748500536729616090625}w^{21}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{897741}{135076700}y^{7}-\frac{4637}{12279700}y^{6}w-\frac{120023}{85311600}y^{6}t+\frac{14428589}{9725522400}y^{5}w^{2}+\frac{250241}{486276120}y^{5}wt+\frac{13293}{540306800}y^{5}t^{2}-\frac{1058171}{29176567200}y^{4}w^{3}-\frac{633}{13507670}y^{4}w^{2}t-\frac{3829}{202615050}y^{4}wt^{2}-\frac{131}{101307525}y^{4}t^{3}+\frac{9197}{911767725}y^{3}w^{4}+\frac{15791}{2735303175}y^{3}w^{3}t+\frac{4831}{2735303175}y^{3}w^{2}t^{2}+\frac{452}{2735303175}y^{3}wt^{3}+\frac{8564}{8205909525}y^{2}w^{5}+\frac{7}{13087575}y^{2}w^{4}t+\frac{236}{2735303175}y^{2}w^{3}t^{2}+\frac{32}{8205909525}y^{2}w^{2}t^{3}-\frac{643}{16411819050}yw^{6}-\frac{52}{2735303175}yw^{5}t-\frac{16}{8205909525}yw^{4}t^{2}+\frac{2}{8205909525}w^{7}$ |
Maps to other modular curves
$j$-invariant map
of degree 30 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{3^3}{2^4}\cdot\frac{3160032826249184xw^{4}+2272862995875416xw^{3}t-488000603151364xw^{2}t^{2}-676665292505111xwt^{3}-128375638260025xt^{4}+17382920063394528yzw^{3}+20412716732559000yzw^{2}t+7678503164938932yzwt^{2}+871542219027645yzt^{3}-7560963699813696yw^{4}-11674870080094656yw^{3}t-2865770784945264yw^{2}t^{2}+1892728077774576ywt^{3}+690702697927920yt^{4}-264342641623040z^{2}w^{3}-500768473259424z^{2}w^{2}t-349204298927344z^{2}wt^{2}-73745479781148z^{2}t^{3}-1209619770254816zw^{4}-411359603116224zw^{3}t+459532562909008zw^{2}t^{2}+229142062848548zwt^{3}+17193614923137zt^{4}+646073609722304w^{5}+714874358650024w^{4}t-797820198972w^{3}t^{2}-279945404971845w^{2}t^{3}-132873099777460wt^{4}-20176226976297t^{5}}{5224957506127xw^{4}+6957244985308xw^{3}t+3259049460358xw^{2}t^{2}+632289733432xwt^{3}+41795818775xt^{4}+23419841023359yzw^{3}+29720624519295yzw^{2}t+12275135606241yzwt^{2}+1634074295085yzt^{3}-12080272909038yw^{4}-24795906057108yw^{3}t-16419488624772yw^{2}t^{2}-4080266148612ywt^{3}-289418850990yt^{4}-356810388340z^{2}w^{3}-676500155412z^{2}w^{2}t-417867520172z^{2}wt^{2}-79433287324z^{2}t^{3}-1956908183968zw^{4}-1785054333747zw^{3}t-318228019771zw^{2}t^{2}+82529383099zwt^{3}+19858321831zt^{4}+844669598887w^{5}+976919924537w^{4}t+66516708834w^{3}t^{2}-312999459210w^{2}t^{3}-144183523705wt^{4}-19858321831t^{5}}$ |
Hi
|
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.
