Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x u - y v - 2 z u + w v $ |
| $=$ | $x v + y u - y v + 2 z u + 2 z v - w u + w v + 3 t u - t v$ |
| $=$ | $2 x u - x v + y v + 2 z u - w u + 4 w v + t u + 2 t v$ |
| $=$ | $x^{2} - 2 x y - x w - x t - y t + 4 z w + 2 z t - w t - t^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 529 x^{12} - 5336 x^{11} z + 154 x^{10} y^{2} + 20586 x^{10} z^{2} - 1526 x^{9} y^{2} z - 44746 x^{9} z^{3} + \cdots + 4 z^{12} $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map
of degree 126 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2\cdot3^2\,\frac{13464618405000118272xt^{10}-116341667837962535808xt^{8}v^{2}+6423038602180676574336xt^{6}v^{4}-954268894185469876200096xt^{4}v^{6}+120330834215425337775532968xt^{2}v^{8}+215568672916098766096316792xv^{10}+66713593288456195200ywt^{9}+1287788564086284536640ywt^{7}v^{2}-19587717894746236516848ywt^{5}v^{4}+3207248091863218469115756ywt^{3}v^{6}-406252357158449338031732895ywtv^{8}+2845615229922705408yt^{10}+663243477734046730752yt^{8}v^{2}-12992951950931110971456yt^{6}v^{4}+2134109606442166650160080yt^{4}v^{6}-268597899980380261033791300yt^{2}v^{8}-135350746282778580162035280yv^{10}+50183180988460869888zwt^{9}+892403940244833781632zwt^{7}v^{2}+11562092369061530746464zwt^{5}v^{4}+12406918691588994904968zwt^{3}v^{6}+2608252845826884064268166zwtv^{8}-250726311354225526848zt^{10}+776787808044940691328zt^{8}v^{2}+6513259383803513736864zt^{6}v^{4}+924247788228451852849656zt^{4}v^{6}-112113990769143973602470310zt^{2}v^{8}+176391610730653234454282304zv^{10}-27418725126706554144w^{2}t^{9}+269040355295828647104w^{2}t^{7}v^{2}+1548356036538063784752w^{2}t^{5}v^{4}-323807775854355574089084w^{2}t^{3}v^{6}+40207264473418642817484435w^{2}tv^{8}+344047252580044604640wt^{10}+2885839711347658455936wt^{8}v^{2}-43603280386500203095104wt^{6}v^{4}+7044249224753648625823056wt^{4}v^{6}-890253997014997831910536836wt^{2}v^{8}-304949897070216865344wu^{10}+34239788374329911808wu^{9}v+1513596964765986505152wu^{8}v^{2}+13301152379671488443712wu^{7}v^{3}+106418551194725537399040wu^{6}v^{4}+1177830414164122806303936wu^{5}v^{5}+3861597983956016287393224wu^{4}v^{6}+30684583132106985007765992wu^{3}v^{7}-147639019302184339875615096wu^{2}v^{8}-34439079053072339271304480wuv^{9}-487947192220500612847314472wv^{10}-7773387945154707456t^{11}+243877888999315946592t^{9}v^{2}-10282055227206830976384t^{7}v^{4}+1611370415898590202779616t^{5}v^{6}-202628125214657670852147096t^{3}v^{8}+1305399982256377206336tu^{10}-464380630203278579328tu^{9}v-9017992734024904776000tu^{8}v^{2}-50756863877165875063104tu^{7}v^{3}-409818408742804788931824tu^{6}v^{4}-4805549132751553649841792tu^{5}v^{5}-14723107071028925900225220tu^{4}v^{6}-119137761034276660960986864tu^{3}v^{7}+416253600988854803675312133tu^{2}v^{8}+554297017239858799776393688tuv^{9}-296016431050660342181949985tv^{10}}{4859440273304607564xt^{8}v^{2}-933094401415194473784xt^{6}v^{4}+368530731917776612655352xt^{4}v^{6}-175789664530219661260646538xt^{2}v^{8}-317282113243297955428364752xv^{10}+668996666273685708ywt^{9}-11087126417570763672ywt^{7}v^{2}+3146612268557776618530ywt^{5}v^{4}-1248344013659346397474569ywt^{3}v^{6}+595684045311265281709429245ywtv^{8}-8427514629916334736yt^{8}v^{2}+2085186665617390336932yt^{6}v^{4}-826302079936398033331146yt^{4}v^{6}+394286240049129245677618521yt^{2}v^{8}+195222509000477485830929808yv^{10}+1046714953865187096zwt^{9}+14906040313258830960zwt^{7}v^{2}+23121733959828490140zwt^{5}v^{4}-35913106214549713756278zwt^{3}v^{6}+18794084479036859649955662zwtv^{8}-2383092720797812758zt^{10}+7218559579862200248zt^{8}v^{2}+938692000713599211492zt^{6}v^{4}-405047357616731677396038zt^{4}v^{6}+195580832294203821635025492zt^{2}v^{8}-258459466704762299636925216zv^{10}-274496035577974371w^{2}t^{9}-539526749309421048w^{2}t^{7}v^{2}-317470284520628731506w^{2}t^{5}v^{4}+131393599543000414846233w^{2}t^{3}v^{6}-62999065389320232357796695w^{2}tv^{8}+3509720984429560845wt^{10}-31890030243077328588wt^{8}v^{2}+6950628687248706401424wt^{6}v^{4}-2756849054314459525215984wt^{4}v^{6}+1316292877382501723496005214wt^{2}v^{8}-11440835607528282726wu^{10}-162862281519580157328wu^{9}v-1670761737720079051626wu^{8}v^{2}-16631783862941909589534wu^{7}v^{3}-135790770080734106187114wu^{6}v^{4}-1144210285871551762614882wu^{5}v^{5}-6416814759197055956741970wu^{4}v^{6}-47708153152163745906614598wu^{3}v^{7}+217915496175162885184422456wu^{2}v^{8}+41620573137122662796963270wuv^{9}+724687922221186777521646004wv^{10}-8169522145146220383t^{9}v^{2}+1576172177489263718952t^{7}v^{4}-622592983510731290848392t^{5}v^{6}+297047685298167309516203340t^{3}v^{8}+48966449218398391014tu^{10}+685136915802405689796tu^{9}v+6884803297515973417884tu^{8}v^{2}+68232490279396263484218tu^{7}v^{3}+552822567885663593712174tu^{6}v^{4}+4650944961162449347813746tu^{5}v^{5}+25550127188270990025994071tu^{4}v^{6}+191005547602514764792134996tu^{3}v^{7}-629591050934442534228245115tu^{2}v^{8}-831982538153866439036293169tuv^{9}+439400136070813574865930251tv^{10}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
28.126.7.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle u$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 7t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle v$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 529X^{12}+154X^{10}Y^{2}+X^{8}Y^{4}-5336X^{11}Z-1526X^{9}Y^{2}Z-14X^{7}Y^{4}Z+20586X^{10}Z^{2}+5936X^{8}Y^{2}Z^{2}+70X^{6}Y^{4}Z^{2}-44746X^{9}Z^{3}-11025X^{7}Y^{2}Z^{3}-189X^{5}Y^{4}Z^{3}+78825X^{8}Z^{4}+7987X^{6}Y^{2}Z^{4}+287X^{4}Y^{4}Z^{4}-100560X^{7}Z^{5}+3528X^{5}Y^{2}Z^{5}-231X^{3}Y^{4}Z^{5}+49021X^{6}Z^{6}-5390X^{4}Y^{2}Z^{6}+56X^{2}Y^{4}Z^{6}-9750X^{5}Z^{7}-4053X^{3}Y^{2}Z^{7}+16XY^{4}Z^{7}-43278X^{4}Z^{8}+5425X^{2}Y^{2}Z^{8}+113692X^{3}Z^{9}+308XY^{2}Z^{9}+63913X^{2}Z^{10}+1004XZ^{11}+4Z^{12} $ |
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.
