Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 30 x^{2} + 15 x y + 16 y^{2} + y z - 2 y w - z w + w^{2} $ |
| $=$ | $15 x y^{2} + x y z - x z w - 16 y^{3} + y^{2} z + 2 y^{2} w + y z^{2} - y z w - y w^{2}$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} y^{2} - 30 x^{4} z^{2} - 2 x^{3} y^{3} - 105 x^{3} y z^{2} - x^{2} y^{4} - 165 x^{2} y^{2} z^{2} + \cdots + 900 y^{2} z^{4} $ |
![Copy content]()
![Toggle raw display]()
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps to other modular curves
$j$-invariant map
of degree 60 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{3^4}{2^4}\cdot\frac{98174104040474459392xyz^{8}-773517362688384172800xyz^{7}w+3136819498956750435840xyz^{6}w^{2}-6449904913403718988800xyz^{5}w^{3}+4093655066273346828000xyz^{4}w^{4}+2000205678312396612000xyz^{3}w^{5}-3139677940022952745500xyz^{2}w^{6}+268444634881348968750xyzw^{7}+332917184672642756250xyw^{8}+14753839145755927808xz^{8}w-174350571844089822720xz^{7}w^{2}+1213555582114503989760xz^{6}w^{3}-3008841944987263641600xz^{5}w^{4}+2873334139543655202000xz^{4}w^{5}-58189546757666277000xz^{3}w^{6}-1188111127025583555750xz^{2}w^{7}+412607358218398342500xzw^{8}-122395831604090601472y^{3}z^{7}-5309953967033160622080y^{3}z^{6}w+33828872139720920770560y^{3}z^{5}w^{2}-64117258326318584256000y^{3}z^{4}w^{3}+27928230664052566992000y^{3}z^{3}w^{4}+46591659304079174280000y^{3}z^{2}w^{5}-49958830049549403582000y^{3}zw^{6}+12695497154514660937500y^{3}w^{7}-91800188603938428928y^{2}z^{8}-186606469373703909376y^{2}z^{7}w+1301509462436977175040y^{2}z^{6}w^{2}-648475077576658072320y^{2}z^{5}w^{3}-3904826301778853436000y^{2}z^{4}w^{4}+6854436034510498230000y^{2}z^{3}w^{5}-5516223122292498067500y^{2}z^{2}w^{6}+2463839988407859305250y^{2}zw^{7}-455286606955374408750y^{2}w^{8}+30322827813706472192yz^{9}-6212060781986356992yz^{8}w-993532282640580289792yz^{7}w^{2}+3696019897398775872000yz^{6}w^{3}-4974170079317595189840yz^{5}w^{4}+1250766024879615705000yz^{4}w^{5}+2371064624926994916750yz^{3}w^{6}-819084920469516760500yz^{2}w^{7}-927922590961869092625yzw^{8}+403098998781636742500yw^{9}+9015333391892480000z^{10}-45076666959462400000z^{9}w+97658235885916053760z^{8}w^{2}-118467117975795475200z^{7}w^{3}+37839947368990285440z^{6}w^{4}+143381434060864752000z^{5}w^{5}-204431947315894761000z^{4}w^{6}+80925610154307259500z^{3}w^{7}+23071088044867017375z^{2}w^{8}-15851419207468891875zw^{9}+950835943676160000w^{10}}{42751977763293049xyz^{8}+145265701506891525xyz^{7}w-195723264682078020xyz^{6}w^{2}-694765753740020475xyz^{5}w^{3}+879021753494166000xyz^{4}w^{4}-723466140749642250xyz^{3}w^{5}+162768468292524000xyz^{2}w^{6}+6054624871453125xyzw^{7}-19289574056081250xyw^{8}-7686700350575899xz^{8}w-7594202848550340xz^{7}w^{2}+186857438798015595xz^{6}w^{3}-242259249831280200xz^{5}w^{4}+176190444036087750xz^{4}w^{5}-30790603699456500xz^{3}w^{6}+18613222996395375xz^{2}w^{7}-5151572370090000xzw^{8}-1286882582669985184y^{3}z^{7}+3374962556583882240y^{3}z^{6}w-1385543753363083680y^{3}z^{5}w^{2}-555458702917032000y^{3}z^{4}w^{3}+3078769854786624000y^{3}z^{3}w^{4}-2661800056908840000y^{3}z^{2}w^{5}+1806276436245396000y^{3}zw^{6}-438988636800000000y^{3}w^{7}-284371368258223616y^{2}z^{8}+1361869599167513828y^{2}z^{7}w-2336339338828563120y^{2}z^{6}w^{2}+1967200139913263460y^{2}z^{5}w^{3}-1130191516838367000y^{2}z^{4}w^{4}-232945351574940000y^{2}z^{3}w^{5}+671311094772015000y^{2}z^{2}w^{6}-458924501659684500y^{2}zw^{7}+99444014459820000y^{2}w^{8}+7686700350575899yz^{9}+2245187658822251yz^{8}w-219959078384771224yz^{7}w^{2}+565977313701944625yz^{6}w^{3}-630171274572222480yz^{5}w^{4}+428610371804291250yz^{4}w^{5}-21643712891694000yz^{3}w^{6}-158903553784102875yz^{2}w^{7}+140748799052724750yzw^{8}-33008094157477500yw^{9}-1168842580423905z^{8}w^{2}+5691678869213100z^{7}w^{3}-36280768395653070z^{6}w^{4}+82456966890172125z^{5}w^{5}-82273720146923250z^{4}w^{6}+26544185655618375z^{3}w^{7}+15530122621082250z^{2}w^{8}-13285275091824375zw^{9}+2785652178738750w^{10}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.60.4.d.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{30}z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -X^{4}Y^{2}-30X^{4}Z^{2}-2X^{3}Y^{3}-105X^{3}YZ^{2}-X^{2}Y^{4}-165X^{2}Y^{2}Z^{2}-120XY^{3}Z^{2}+900XYZ^{4}-60Y^{4}Z^{2}+900Y^{2}Z^{4} $ |
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.
