Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} - x w + x t + y t $ |
| $=$ | $x^{2} + x z + 2 x w + y t - z^{2} - 2 z w - w^{2}$ |
| $=$ | $x^{2} + y^{2} + 2 y z + 2 y w - y t + z^{2} + 2 z w - z t$ |
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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 |
|---|
| $(0:0:-1:1:1)$, $(1/3:-1/6:-1/6:5/6:1)$, $(0:1:-1:1:0)$, $(1:-2:1:1:0)$, $(-1/3:1/6:1/6:1/6:1)$, $(1:1:-2:1:0)$, $(0:-1:-1:1:0)$, $(0:0:0:0:1)$ |
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{-1101204000yz^{2}w^{9}+769937616xyw^{10}-2473473600yzw^{10}-2076263568z^{2}w^{10}+1805178224xw^{11}-1312033824yw^{11}-4363352352zw^{11}-2226851984w^{12}+3375732240yz^{2}w^{8}t-7481495504xyw^{9}t+5059358928yzw^{9}t-2224674576z^{2}w^{9}t+637202672xw^{10}t+4332162928yw^{10}t-2866362624zw^{10}t+1573843584w^{11}t+40513615200yz^{2}w^{7}t^{2}-41986389168xyw^{8}t^{2}+90289768464yzw^{8}t^{2}+71611914960z^{2}w^{8}t^{2}-66658243360xw^{9}t^{2}+50922213632yw^{9}t^{2}+141552055152zw^{9}t^{2}+64696335456w^{10}t^{2}-84131529840yz^{2}w^{6}t^{3}+94680843888xyw^{7}t^{3}-184954443408yzw^{7}t^{3}-102846908976z^{2}w^{7}t^{3}+139134320784xw^{8}t^{3}-179961485232yw^{8}t^{3}-229062506496zw^{8}t^{3}-149322391280w^{9}t^{3}-65404144400yz^{2}w^{5}t^{4}-64657507936xyw^{6}t^{4}-275538610864yzw^{6}t^{4}-404471962768z^{2}w^{6}t^{4}+283872442144xw^{7}t^{4}-138716908928yw^{7}t^{4}-728395806992zw^{7}t^{4}-189564572464w^{8}t^{4}+349910930960yz^{2}w^{4}t^{5}-230210553200xyw^{5}t^{5}+1098616893328yzw^{5}t^{5}+1042142507552z^{2}w^{5}t^{5}-1124585022912xw^{6}t^{5}+1174192396032yw^{6}t^{5}+2060241092688zw^{6}t^{5}+863050363056w^{7}t^{5}-306811384560yz^{2}w^{3}t^{6}+444699730512xyw^{4}t^{6}-930701159904yzw^{4}t^{6}-704274317872z^{2}w^{4}t^{6}+1204755984192xw^{5}t^{6}-1482336677680yw^{5}t^{6}-1486108944816zw^{5}t^{6}-876855455296w^{6}t^{6}+55178056000yz^{2}w^{2}t^{7}-290351294784xyw^{3}t^{7}-66344877376yzw^{3}t^{7}-226399477392z^{2}w^{3}t^{7}-342853547984xw^{4}t^{7}+344197675120yw^{4}t^{7}-410738278480zw^{4}t^{7}+177772535440w^{5}t^{7}+32079604400yz^{2}wt^{8}+86979125168xyw^{2}t^{8}+419469380720yzw^{2}t^{8}+490794373216z^{2}w^{2}t^{8}-285842029152xw^{3}t^{8}+606409820608yw^{3}t^{8}+1057266016096zw^{3}t^{8}+221840899680w^{4}t^{8}-10173090480yz^{2}t^{9}-10629427568xywt^{9}-178987210096yzwt^{9}-208058806640z^{2}wt^{9}+265598679184xw^{2}t^{9}-498748197968yw^{2}t^{9}-495548508720zw^{2}t^{9}-135234251696w^{3}t^{9}-126594064xyt^{10}+21875600944yzt^{10}+27703743328z^{2}t^{10}-83137846432xwt^{10}+140392458576ywt^{10}+78803719088zwt^{10}+21413254944w^{2}t^{10}+9492339776xt^{11}-13658082080yt^{11}-1737168448zt^{11}-240wt^{11}+16t^{12}}{-4448yz^{2}w^{8}t+17536xyw^{9}t-8768yzw^{9}t+3152z^{2}w^{9}t-12352xw^{10}t-4320yw^{10}t+6176zw^{10}t+3024w^{11}t+59552yz^{2}w^{7}t^{2}-246848xyw^{8}t^{2}+129072yzw^{8}t^{2}-55600z^{2}w^{8}t^{2}+247232xw^{9}t^{2}+74672yw^{9}t^{2}-103616zw^{9}t^{2}-55504w^{10}t^{2}-132640yz^{2}w^{6}t^{3}+627936xyw^{7}t^{3}-386464yzw^{7}t^{3}+268176z^{2}w^{7}t^{3}-1524176xw^{8}t^{3}-375024yw^{8}t^{3}+459440zw^{8}t^{3}+288992w^{9}t^{3}-799216yz^{2}w^{5}t^{4}+3643248xyw^{6}t^{4}-1642000yzw^{6}t^{4}+5936z^{2}w^{6}t^{4}+1846592xw^{7}t^{4}-19280yw^{7}t^{4}+182112zw^{7}t^{4}-8896w^{8}t^{4}+2680744yz^{2}w^{4}t^{5}-15969488xyw^{5}t^{5}+8393512yzw^{5}t^{5}-2555496z^{2}w^{5}t^{5}+11289696xw^{6}t^{5}+4655040yw^{6}t^{5}-4204664zw^{6}t^{5}-3263024w^{7}t^{5}-509900yz^{2}w^{3}t^{6}+9835120xyw^{4}t^{6}-4802860yzw^{4}t^{6}+5910668z^{2}w^{4}t^{6}-33941088xw^{5}t^{6}-10263984yw^{5}t^{6}+6877964zw^{5}t^{6}+6473072w^{6}t^{6}-618744yz^{2}w^{2}t^{7}+9024300xyw^{3}t^{7}-11762024yzw^{3}t^{7}-10781672z^{2}w^{3}t^{7}+29636740xw^{4}t^{7}+2007608yw^{4}t^{7}-13971664zw^{4}t^{7}-4031968w^{5}t^{7}-1133694yz^{2}wt^{8}-1489758xyw^{2}t^{8}+12120974yzw^{2}t^{8}+14161908z^{2}w^{2}t^{8}-10137902xw^{3}t^{8}+17107190yw^{3}t^{8}+27609696zw^{3}t^{8}+3293846w^{4}t^{8}+559595yz^{2}t^{9}-3110813xywt^{9}-6953923yzwt^{9}-9022495z^{2}wt^{9}+1087587xw^{2}t^{9}-22679769yw^{2}t^{9}-22311507zw^{2}t^{9}-2425095w^{3}t^{9}+745595xyt^{10}+2100788yzt^{10}+2217743z^{2}t^{10}-518593xwt^{10}+9341088ywt^{10}+7258392zwt^{10}+673864w^{2}t^{10}+176194xt^{11}-1199368yt^{11}-701698zt^{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.
