Canonical model in $\mathbb{P}^{ 5 }$ defined by 6 equations
| $ 0 $ | $=$ | $ 2 x w - 2 x t + 2 x u + 2 y z - y w + y t + z w + w^{2} - w t - 2 w u $ |
| $=$ | $x^{2} - 2 x y - x w + 2 x t - x u - 2 y w - 4 y t + 2 y u - z w + w t$ |
| $=$ | $x y - x z - 2 x w + x t - x u - 2 y^{2} - y w + y t - 2 y u - 3 z w + 2 w t - w u$ |
| $=$ | $x^{2} - x w + 2 x u - 4 y^{2} - 3 y z + y t - 3 y u - z^{2} + z w + z t - z u - w t - w u - t^{2} + \cdots + 2 u^{2}$ |
| $=$ | $\cdots$ |
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This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve |
CM |
$j$-invariant |
$j$-height | Canonical model |
| 27.a3 |
$-3$ | $0$ |
| $0.000$ | $(-1:-1/2:-1/2:0:1:1)$, $(-8/9:-7/27:-13/27:2/27:34/27:1)$, $(-3/2:1/2:-1:1/2:1:0)$ |
|
no | $\infty$ |
| $0.000$ | $(2:1:0:0:0:0)$, $(0:-1:0:1:0:1)$ |
| 36.a1 |
$-12$ | $54000$ |
$= 2^{4} \cdot 3^{3} \cdot 5^{3}$ | $10.897$ | $(2:1:-5:0:-2:1)$ |
| 27.a1 |
$-27$ | $-12288000$ |
$= -1 \cdot 2^{15} \cdot 3 \cdot 5^{3}$ | $16.324$ | $(1:1/3:-2/3:4/3:2/3:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 112 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{-370063318371571714003623936y^{12}+5304632452040841086432467337072y^{2}t^{10}+304060744719766105482499082428ywt^{10}+2478526581420113960440321174572zwt^{10}-1019476455726020569770297421820w^{2}t^{10}-4077137377662230207122348213900yt^{11}-2099346859464706225529341383512zt^{11}-1085780650438587765667287420628wt^{11}+574071350384602922358970260784t^{12}-1560972995837487036914651136y^{11}u+7144034061341788725748411948856y^{2}t^{9}u+98095575405045761876393747428ywt^{9}u+2308567443519375284601340499724zwt^{9}u-898591118373096612903603236996w^{2}t^{9}u-685960072658510786360045386992xt^{10}u+994447912845958187957300423484yt^{10}u-1034695974290709006935257736004zt^{10}u+816702506582164597950911537832wt^{10}u-1902654784342606442066889652668t^{11}u-2798253577632815229704491008y^{10}u^{2}+2676417085700997227728836567976y^{2}t^{8}u^{2}-2550336802050745155197320804296ywt^{8}u^{2}-1200419935740139458370878067720zwt^{8}u^{2}+398406381579179116779247731672w^{2}t^{8}u^{2}-494639114971353208685612900340xt^{9}u^{2}+7897223621784414128919221373636yt^{9}u^{2}+2307669289813122128641273672664zt^{9}u^{2}+2298906095633356176603100942904wt^{9}u^{2}-1012299842568732628050566395228t^{10}u^{2}-434376960536049797665953792y^{9}u^{3}-2307699430476670842515400665728y^{2}t^{7}u^{3}-4807973095691879495289414283280ywt^{7}u^{3}-3387235220278698518966246683456zwt^{7}u^{3}+1552909499744434181784420554064w^{2}t^{7}u^{3}+1093678226106759057827470521540xt^{8}u^{3}+2833561884772583392974162117340yt^{8}u^{3}+2253669179015208713332938151060zt^{8}u^{3}+1578773155651539934414476060052wt^{8}u^{3}+2186435759910029550112869938784t^{9}u^{3}+6938201364475214093366002176y^{8}u^{4}+6693521770920862708594615465488y^{2}t^{6}u^{4}+5620395783011083915419800013660ywt^{6}u^{4}+6874864668572764034218259732044zwt^{6}u^{4}-1725960430741479047152216327708w^{2}t^{6}u^{4}+382285571806912108402689161288xt^{7}u^{4}-15212373005902379584515981230740yt^{7}u^{4}-5741021037763088638377826694384zt^{7}u^{4}-4871646180853552014231982157836wt^{7}u^{4}+3631903729549847867979014450488t^{8}u^{4}+12626874081781541623827975168y^{7}u^{5}+700788782919033736442508101416y^{2}t^{5}u^{5}+4224582784963117102403382931244ywt^{5}u^{5}+1216971808463825194186374098228zwt^{5}u^{5}-746300334220850791086492346524w^{2}t^{5}u^{5}-2944321899586655766716435064496xt^{6}u^{5}+16688717070418620999792741628236yt^{6}u^{5}+3713189018091831131009035658016zt^{6}u^{5}+2571055495223976958235785461820wt^{6}u^{5}-10510481492377444763079237121296t^{7}u^{5}+12097990557318647038496453280y^{6}u^{6}-3110114846979923141229435147296y^{2}t^{4}u^{6}-3914137972912377655692626150569ywt^{4}u^{6}-4472958774021738013783932432683zwt^{4}u^{6}+876043849358052979318646711297w^{2}t^{4}u^{6}+2693831385226048639010118695792xt^{5}u^{6}+784617342040864139377441007815yt^{5}u^{6}+2042102762879023031150267850308zt^{5}u^{6}+2932006871819973745290994357689wt^{5}u^{6}+4774511728735280925545212851728t^{6}u^{6}+13720207410946831126157937216y^{5}u^{7}+1079539929296353687702066478224y^{2}t^{3}u^{7}+23808156254209580281290729740ywt^{3}u^{7}+1234112822137602576495761470980zwt^{3}u^{7}-277705618212382200626755422628w^{2}t^{3}u^{7}+501098195127019461713903104356xt^{4}u^{7}-7517918249908195961296299135336yt^{4}u^{7}-1913497494519937277451570233338zt^{4}u^{7}-2553316158800386176399969147178wt^{4}u^{7}+3221833509290319945695240198604t^{5}u^{7}+27379172729564417801656984752y^{4}u^{8}+98625982721720690349135195726y^{2}t^{2}u^{8}+251271262682241391747064619355ywt^{2}u^{8}+251559634233247184110217380597zwt^{2}u^{8}-27371010417300194051005497171w^{2}t^{2}u^{8}-671609155150391158459987082162xt^{3}u^{8}+2607313042671698448434306475803yt^{3}u^{8}+343969134066605980139654753712zt^{3}u^{8}+751915207220169784635030373893wt^{3}u^{8}-3913386265060414742473674156482t^{4}u^{8}+56478059303565351137603475348y^{3}u^{9}-103161113829111727627334931324y^{2}tu^{9}+23715527477330339470440099836ywtu^{9}-19263973201907835989726293876zwtu^{9}+9118351079568334278060266412w^{2}tu^{9}+77472876069469431999705140562xt^{2}u^{9}+195976265729770208365131898862yt^{2}u^{9}+59869425504233725280982457397zt^{2}u^{9}+100104626689431585103966441309wt^{2}u^{9}+771127239032073119241824916613t^{3}u^{9}+61623532429670029159516936838y^{2}u^{10}+47571478534481168837907007216ywu^{10}+62012794411022770990493036022zwu^{10}-31644186986153705273724094072w^{2}u^{10}+43217887726969938102926304275xtu^{10}-148056466968795579005604447389ytu^{10}-58348812326022259386283389948ztu^{10}-55304791549064687238229142494wtu^{10}+144728887861111706841717249081t^{2}u^{10}+4946100303732283165267506177xu^{11}+38909584270984328590474029209yu^{11}+2136192149980604789250639147zu^{11}+57544607428431169235235926999wu^{11}-75731951665868619178502629870tu^{11}+36539345386646203936099787816u^{12}}{-12941283110457696164958256y^{2}t^{10}-1926438396143366835146998ywt^{10}-6518226477959464955265334zwt^{10}+2657048526105731441982746w^{2}t^{10}+10961483049488392061114978yt^{11}+5474517282198027725042808zt^{11}+28625201033683246741414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|
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.
