Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x z w + y w^{2} $ |
| $=$ | $x z^{2} + y z w$ |
| $=$ | $x y z + y^{2} w$ |
| $=$ | $x^{2} z + x y w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} - 2 x^{4} z - x^{3} z^{2} + 6 x^{2} y^{2} z + 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\cdot3^2}\cdot\frac{30233088x^{2}y^{18}+105667101517824x^{2}y^{16}w^{2}+98741949011712x^{2}y^{14}w^{4}+17708435312256x^{2}y^{12}w^{6}+6926054553216x^{2}y^{10}w^{8}-17562886352640x^{2}y^{8}w^{10}+23721996407256x^{2}y^{6}w^{12}-15180428223504x^{2}y^{4}w^{14}+4529244339882x^{2}y^{2}w^{16}-508236398592x^{2}w^{18}-63403380965376xy^{19}-116238710437632xy^{17}w^{2}-7064330526720xy^{15}w^{4}+17838274807872xy^{13}w^{6}+1027592428032xy^{11}w^{8}+5269753194048xy^{9}w^{10}-7997793391296xy^{7}w^{12}+5608405221192xy^{5}w^{14}-1786307937840xy^{3}w^{16}+210518409213xyw^{18}+30233088y^{20}+10567129383936y^{18}z^{2}-21135291731712y^{18}zw-21139751112192y^{18}w^{2}+15855909283584y^{16}z^{2}w^{2}-17591108815872y^{16}zw^{3}-56241004534272y^{16}w^{4}+1078506067968y^{14}z^{2}w^{4}+6787754458560y^{14}zw^{5}-22264361799552y^{14}w^{6}-145857643488y^{12}z^{2}w^{6}+4967282874816y^{12}zw^{7}+4996248342624y^{12}w^{8}-3938250129408y^{10}z^{2}w^{8}-6389035144128y^{10}zw^{9}-19054116964224y^{10}w^{10}+9652208442552y^{8}z^{2}w^{10}+11795325738048y^{8}zw^{11}+26377422195600y^{8}w^{12}-11567874928320y^{6}z^{2}w^{12}-10310839612200y^{6}zw^{13}-17237880050544y^{6}w^{14}+6805417611348y^{4}z^{2}w^{14}+4592058259752y^{4}zw^{15}+5231974825062y^{4}w^{16}-1909626765300y^{2}z^{2}w^{16}-1001511714843y^{2}zw^{17}-595435978758y^{2}w^{18}+204498534400z^{2}w^{18}+84706066432zw^{19}}{w^{2}y^{4}(6018624x^{2}y^{10}w^{2}-48335616x^{2}y^{8}w^{4}+55331424x^{2}y^{6}w^{6}-14580864x^{2}y^{4}w^{8}-3024x^{2}y^{2}w^{10}-36x^{2}w^{12}-746496xy^{11}w^{2}+13654656xy^{9}w^{4}-20331648xy^{7}w^{6}+6038064xy^{5}w^{8}-576xy^{3}w^{10}+6xyw^{12}+46656y^{12}z^{2}+746496y^{12}zw+6065280y^{12}w^{2}-5723136y^{10}z^{2}w^{2}-15396480y^{10}zw^{3}-51819264y^{10}w^{4}+26671680y^{8}z^{2}w^{4}+31010688y^{8}zw^{5}+62708256y^{8}w^{6}-24772608y^{6}z^{2}w^{6}-16299792y^{6}zw^{7}-17095968y^{6}w^{8}+5869008y^{4}z^{2}w^{8}+2428992y^{4}zw^{9}+2016y^{4}w^{10}-96y^{2}z^{2}w^{10}+90y^{2}zw^{11}+348y^{2}w^{12}-z^{2}w^{12}+2zw^{13}+w^{14})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
48.96.2.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{5}-2X^{4}Z+6X^{2}Y^{2}Z-X^{3}Z^{2}+6Y^{2}Z^{3} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
48.96.2.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -6yz^{4}w-6yz^{2}w^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle zw$ |
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.
