Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} + 5 x y - 2 x z + 2 z^{2} $ |
| $=$ | $12 x^{2} - 5 x y + 2 x z - 15 y^{2} - 2 z^{2} + w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 3 x^{3} z + 15 x^{2} y^{2} - 4 x^{2} z^{2} + 2 x z^{3} - z^{4} $ |
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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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{6}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^6\cdot5^2}\cdot\frac{536824589037879353906250xz^{17}-112071461623369885546875xz^{15}w^{2}+8847971165275878656250xz^{13}w^{4}-232550017691349206250xz^{11}w^{6}-9054432007905510000xz^{9}w^{8}+617320634518199250xz^{7}w^{10}-1808807323261500xz^{5}w^{12}-495754811945820xz^{3}w^{14}+3210538865400xzw^{16}-843809916603264454687500y^{2}z^{16}+119633754909684979687500y^{2}z^{14}w^{2}-6639426709631550562500y^{2}z^{12}w^{4}+11638220932270575000y^{2}z^{10}w^{6}+10913180827348327500y^{2}z^{8}w^{8}-315363363010137000y^{2}z^{6}w^{10}-8093987727831000y^{2}z^{4}w^{12}+182993203705680y^{2}z^{2}w^{14}-239753397900y^{2}w^{16}-325973165343560191406250yz^{17}+38696131409236607812500yz^{15}w^{2}-1750874631110524687500yz^{13}w^{4}-27253908859156875000yz^{11}w^{6}+3279283523413537500yz^{9}w^{8}-58571665808895000yz^{7}w^{10}-2852044574661000yz^{5}w^{12}+19263233192400yz^{3}w^{14}-210931743831167545312500z^{18}+79440238650301939687500z^{16}w^{2}-8950685158373500687500z^{14}w^{4}+420389686450080337500z^{12}w^{6}+1065134042961930000z^{10}w^{8}-756566051406826500z^{8}w^{10}+18080257642201800z^{6}w^{12}+547879883287320z^{4}w^{14}-12199546913712z^{2}w^{16}-1083528067916w^{18}}{w^{6}(86509462434750xz^{11}-12758815984725xz^{9}w^{2}+554761168110xz^{7}w^{4}-8441345745xz^{5}w^{6}+40698450xz^{3}w^{8}-41625xzw^{10}-136328116486500y^{2}z^{10}+10855842579900y^{2}z^{8}w^{2}-269459544240y^{2}z^{6}w^{4}+2269185300y^{2}z^{4}w^{6}-5291100y^{2}z^{2}w^{8}+1500y^{2}w^{10}-52350706518750yz^{11}+2973751866000yz^{9}w^{2}-49043244150yz^{7}w^{4}+242649000yz^{5}w^{6}-249750yz^{3}w^{8}-33753532153500z^{12}+10734797378700z^{10}w^{2}-747968119920z^{8}w^{4}+18072974016z^{6}w^{6}-151382520z^{4}w^{8}+352740z^{2}w^{10}-100w^{12})}$ |
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.
