Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x w + x t + y z $ |
| $=$ | $x^{2} - x z + y^{2} - y w - 2 w^{2} - w t$ |
| $=$ | $2 x^{2} + x z - y^{2} - y w - y t + z^{2}$ |
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This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:-1/2:1)$, $(1/4:1/4:-1/4:-3/4:1)$, $(0:-1:0:1:0)$, $(1:1:-1:1:0)$, $(-1:1:1:1:0)$, $(0:0:0:0:1)$, $(-1/4:1/4:1/4:-3/4:1)$, $(0:-1/2:0:-1/2:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1135037367005184xzw^{10}-28255802600448z^{2}w^{10}+1253899280553984yw^{11}+1275240635974656w^{12}+2036020931103744xzw^{9}t-1365623959104000z^{2}w^{9}t+3990798652580352yw^{10}t+4699935450602496w^{11}t-1406920958198784xzw^{8}t^{2}-4122187476431616z^{2}w^{8}t^{2}+5678214215083776yw^{9}t^{2}+7367525253024768w^{10}t^{2}-7370514036806400xzw^{7}t^{3}-5026625362870272z^{2}w^{7}t^{3}+4409011047301632yw^{8}t^{3}+5272062659039232w^{9}t^{3}-9507922079842368xzw^{6}t^{4}-2839847559032448z^{2}w^{6}t^{4}+1163430481374144yw^{7}t^{4}-635010070352256w^{8}t^{4}-6684854404309632xzw^{5}t^{5}-362268602204832z^{2}w^{5}t^{5}-1420426190414304yw^{6}t^{5}-4936613000368320w^{7}t^{5}-2865912521759424xzw^{4}t^{6}+521087837456880z^{2}w^{4}t^{6}-1955027882353968yw^{5}t^{6}-4870860476360832w^{6}t^{6}-747421848988992xzw^{3}t^{7}+391756308491376z^{2}w^{3}t^{7}-1269850300959456yw^{4}t^{7}-2650862121711840w^{5}t^{7}-115410029949036xzw^{2}t^{8}+140460606433224z^{2}w^{2}t^{8}-547895847283884yw^{3}t^{8}-935591772158136w^{4}t^{8}-12959901230868xzwt^{9}+27564020764758z^{2}wt^{9}-163713754657602yw^{2}t^{9}-227777268893724w^{3}t^{9}-1486744216470xzt^{10}+2325610613313z^{2}t^{10}-29826873505575ywt^{10}-37033684449552w^{2}t^{10}-2324522222145yt^{11}-3064901254668wt^{11}+181398528t^{12}}{941270482944xzw^{10}+292294541312z^{2}w^{10}+324487970816yw^{11}+324487970816w^{12}+5201778941952xzw^{9}t+1991395581952z^{2}w^{9}t+1626653966336yw^{10}t+1940410793984w^{11}t+12806421746688xzw^{8}t^{2}+5942619436032z^{2}w^{8}t^{2}+2872520443904yw^{9}t^{2}+4747682760704w^{10}t^{2}+17996144451840xzw^{7}t^{3}+10289105285376z^{2}w^{7}t^{3}+651195904512yw^{8}t^{3}+5540038288896w^{9}t^{3}+15342136201728xzw^{6}t^{4}+11357012435712z^{2}w^{6}t^{4}-5497549679616yw^{7}t^{4}+1645429101312w^{8}t^{4}+7764586159104xzw^{5}t^{5}+8290962588672z^{2}w^{5}t^{5}-10026239396352yw^{6}t^{5}-3802929605376w^{7}t^{5}+1959215986368xzw^{4}t^{6}+4037969275584z^{2}w^{4}t^{6}-8889496879296yw^{5}t^{6}-5742399592704w^{6}t^{6}-37421197560xzw^{3}t^{7}+1292891783664z^{2}w^{3}t^{7}-4707342333048yw^{4}t^{7}-3968106570480w^{5}t^{7}-167614659612xzw^{2}t^{8}+259968528252z^{2}w^{2}t^{8}-1546038973488yw^{3}t^{8}-1613295344928w^{4}t^{8}-42158813256xzwt^{9}+29540722708z^{2}wt^{9}-306131251388yw^{2}t^{9}-395436067700w^{3}t^{9}-3571117746xzt^{10}+1430023595z^{2}t^{10}-32934072815ywt^{10}-54281193800w^{2}t^{10}-1430023595yt^{11}-3215582468wt^{11}}$ |
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.
