Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x z w - x w^{2} - y w^{2} $ |
| $=$ | $x z^{2} - x z w - y z w$ |
| $=$ | $x^{2} z - x^{2} w - x y w$ |
| $=$ | $x y z - x y w - y^{2} w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} - 4 x^{3} z^{2} + 6 x^{2} y^{2} z + 4 x^{2} z^{3} - x z^{4} + 6 y^{2} z^{3} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -6x^{5} - 12x^{4} - 12x^{2} + 6x $ |
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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 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^4\cdot3^3}\cdot\frac{61947597312x^{2}y^{18}+557387288064x^{2}y^{16}w^{2}-1343916188928x^{2}y^{14}w^{4}+1072305625536x^{2}y^{12}w^{6}+92794328352x^{2}y^{10}w^{8}+559336272480x^{2}y^{8}w^{10}+939758038776x^{2}y^{6}w^{12}+796791469644x^{2}y^{4}w^{14}+349790183388x^{2}y^{2}w^{16}+61659340800x^{2}w^{18}+123864961536xy^{19}-82475864064xy^{17}w^{2}-489821375232xy^{15}w^{4}+481520433024xy^{13}w^{6}+477646468704xy^{11}w^{8}+356488231680xy^{9}w^{10}+695199470472xy^{7}w^{12}+576836370648xy^{5}w^{14}+250112286720xy^{3}w^{16}+43598807040xyw^{18}+15116544y^{20}+20639121408y^{18}z^{2}+30928449024y^{18}zw-61882092288y^{18}w^{2}-9999593856y^{16}z^{2}w^{2}+69279121152y^{16}zw^{3}+223195772160y^{16}w^{4}-62621963136y^{14}z^{2}w^{4}-277601263776y^{14}zw^{5}-235241278272y^{14}w^{6}+54488317680y^{12}z^{2}w^{6}+164752693632y^{12}zw^{7}+422329412160y^{12}w^{8}+83020024656y^{10}z^{2}w^{8}-101662374240y^{10}zw^{9}+195912719712y^{10}w^{10}+202372175844y^{8}z^{2}w^{10}-204477441336y^{8}zw^{11}+390306942792y^{8}w^{12}+295685175924y^{6}z^{2}w^{12}-231615279414y^{6}zw^{13}+329596016868y^{6}w^{14}+240884748099y^{4}z^{2}w^{14}-149316452748y^{4}zw^{15}+144926539929y^{4}w^{16}+102160281597y^{2}z^{2}w^{16}-51347324934y^{2}zw^{17}+25539256323y^{2}w^{18}+17543200768z^{2}w^{18}-7266631680zw^{19}}{w^{2}y^{4}(13608x^{2}y^{10}w^{2}+54756x^{2}y^{8}w^{4}+83700x^{2}y^{6}w^{6}+57888x^{2}y^{4}w^{8}+14592x^{2}y^{2}w^{10}-384x^{2}w^{12}+17496xy^{11}w^{2}+73224xy^{9}w^{4}+114912xy^{7}w^{6}+78336xy^{5}w^{8}+16128xy^{3}w^{10}-2816xyw^{12}+972y^{12}z^{2}+1944y^{12}zw+9720y^{12}w^{2}+4860y^{10}z^{2}w^{2}+7614y^{10}zw^{3}+39852y^{10}w^{4}+7641y^{8}z^{2}w^{4}+9180y^{8}zw^{5}+60075y^{8}w^{6}+3591y^{6}z^{2}w^{6}+558y^{6}zw^{7}+37737y^{6}w^{8}-960y^{4}z^{2}w^{8}-5568y^{4}zw^{9}+5376y^{4}w^{10}-832y^{2}z^{2}w^{10}-2688y^{2}zw^{11}-2432y^{2}w^{12})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
48.96.2.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{5}+6X^{2}Y^{2}Z-4X^{3}Z^{2}+4X^{2}Z^{3}+6Y^{2}Z^{3}-XZ^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
48.96.2.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z^{2}-zw$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -6yz^{4}w+12yz^{3}w^{2}-12yz^{2}w^{3}+12yzw^{4}-6yw^{5}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle zw-w^{2}$ |
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.
