Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + 2 x w - y^{2} + 2 y z + 2 z^{2} $ |
| $=$ | $4 x^{2} + 4 x y + 8 x z - 3 y^{2} + 2 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} + 24 x^{3} y - 4 x^{3} z + 36 x^{2} y^{2} - 24 x^{2} y z - 2 x^{2} z^{2} - 48 x y^{3} + \cdots + 9 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 32 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\cdot3\,\frac{1274632028928xz^{7}-8374614411456xz^{6}w+23189512581504xz^{5}w^{2}-35271156620160xz^{4}w^{3}+31907385996480xz^{3}w^{4}-17180959058736xz^{2}w^{5}+5096623226592xzw^{6}-642033979248xw^{7}+880308755568y^{2}z^{6}-4183237367856y^{2}z^{5}w+8335991959380y^{2}z^{4}w^{2}-8900800998480y^{2}z^{3}w^{3}+5359241715120y^{2}z^{2}w^{4}-1721568034956y^{2}zw^{5}+229987696357y^{2}w^{6}-2328111678528yz^{7}+11090378523456yz^{6}w-22256151393312yz^{5}w^{2}+24116048725680yz^{4}w^{3}-14959668455280yz^{3}w^{4}+5125951106496yz^{2}w^{5}-823340251832yzw^{6}+32599222924yw^{7}-2614567143024z^{8}+12245472608256z^{7}w-23553310377600z^{6}w^{2}+23180331710592z^{5}w^{3}-11249454294000z^{4}w^{4}+1230911859648z^{3}w^{5}+1196624115928z^{2}w^{6}-490922008960zw^{7}+53238660231w^{8}}{10576751904xz^{7}-66149556228xz^{6}w+174917548872xz^{5}w^{2}-255870077040xz^{4}w^{3}+224738381520xz^{3}w^{4}-118890808188xz^{2}w^{5}+35149638216xzw^{6}-4487368224xw^{7}+7304912424y^{2}z^{6}-32414771403y^{2}z^{5}w+61268046840y^{2}z^{4}w^{2}-62994290490y^{2}z^{3}w^{3}+37108480860y^{2}z^{2}w^{4}-11863726983y^{2}zw^{5}+1607341516y^{2}w^{6}-19317972204yz^{7}+85951018728yz^{6}w-163697171616yz^{5}w^{2}+170936010720yz^{4}w^{3}-103766133780yz^{3}w^{4}+35353560648yz^{2}w^{5}-5729540576yzw^{6}+227954512yw^{7}-21695173407z^{8}+94783990968z^{7}w-172427585220z^{6}w^{2}+162631750266z^{5}w^{3}-76545143910z^{4}w^{4}+8018351844z^{3}w^{5}+8216160844z^{2}w^{6}-3391062670zw^{7}+372237753w^{8}}$ |
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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.
