Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x w^{2} + x w t + y z w + y w^{2} + y w t $ |
| $=$ | $x z w + x z t + y z^{2} + y z w + y z t$ |
| $=$ | $x w t + x t^{2} + y z t + y w t + y t^{2}$ |
| $=$ | $x^{2} w + x^{2} t + x y z - y^{2} z - y^{2} w - y^{2} t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 8 x^{6} z + 24 x^{5} y^{2} - 20 x^{5} z^{2} + 72 x^{4} y^{2} z - 24 x^{4} z^{3} + 48 x^{3} y^{2} z^{2} + \cdots - 6 y^{2} z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -6x^{7} + 42x^{5} - 42x^{3} + 6x $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:1:0:1)$, $(0:-2:-1:1:0)$, $(2:0:1:0:0)$, $(2:-2:0:1:0)$ |
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}{3}\cdot\frac{55594357xyt^{12}-139968xw^{13}-2426112xw^{12}t-2939328xw^{11}t^{2}-33778944xw^{10}t^{3}-57670704xw^{9}t^{4}+143570880xw^{8}t^{5}+287802720xw^{7}t^{6}-360033120xw^{6}t^{7}-1082232036xw^{5}t^{8}-246178512xw^{4}t^{9}+1023888564xw^{3}t^{10}+1029317664xw^{2}t^{11}+394390828xwt^{12}+26567712xt^{13}+3079296y^{2}w^{12}+9517824y^{2}w^{11}t+41928192y^{2}w^{10}t^{2}+95769216y^{2}w^{9}t^{3}+114942240y^{2}w^{8}t^{4}-28781568y^{2}w^{7}t^{5}-453231936y^{2}w^{6}t^{6}-535410432y^{2}w^{5}t^{7}-80997480y^{2}w^{4}t^{8}+578666736y^{2}w^{3}t^{9}+568567872y^{2}w^{2}t^{10}+233365032y^{2}wt^{11}+144726104y^{2}t^{12}+3079296yw^{13}+5085504yw^{12}t+21492864yw^{11}t^{2}-162254016yw^{10}t^{3}-494416224yw^{9}t^{4}-231345072yw^{8}t^{5}+703935360yw^{7}t^{6}+1359891936yw^{6}t^{7}+598151448yw^{5}t^{8}-298360260yw^{4}t^{9}-1288138104yw^{3}t^{10}-1414968924yw^{2}t^{11}-680506528ywt^{12}-339077592yt^{13}+1119744z^{14}-1306368z^{12}t^{2}+5225472z^{11}t^{3}+995328z^{10}t^{4}-3483648z^{9}t^{5}+7776000z^{8}t^{6}+1140480z^{7}t^{7}-1436832z^{6}t^{8}+4278528z^{5}t^{9}-329904z^{4}t^{10}+1330560z^{3}t^{11}-10217664z^{2}w^{12}-15116544z^{2}w^{11}t-83187648z^{2}w^{10}t^{2}-388924416z^{2}w^{9}t^{3}-705053808z^{2}w^{8}t^{4}+200029824z^{2}w^{7}t^{5}+2346198048z^{2}w^{6}t^{6}+2558972736z^{2}w^{5}t^{7}-34946964z^{2}w^{4}t^{8}-2765250576z^{2}w^{3}t^{9}-2916738684z^{2}w^{2}t^{10}-1470949056z^{2}wt^{11}-477044784z^{2}t^{12}+139968zw^{13}+3032640zw^{12}t+7581600zw^{11}t^{2}+29276640zw^{10}t^{3}-30944592zw^{9}t^{4}-46307376zw^{8}t^{5}+325621944zw^{7}t^{6}+417326472zw^{6}t^{7}-392405796zw^{5}t^{8}-1056731148zw^{4}t^{9}-285403050zw^{3}t^{10}+312174498zw^{2}t^{11}+202073684zwt^{12}+55956700zt^{13}-5878656w^{14}-4852224w^{13}t-70209504w^{12}t^{2}-389017728w^{11}t^{3}-491126976w^{10}t^{4}+820596096w^{9}t^{5}+2622420360w^{8}t^{6}+1389014784w^{7}t^{7}-2564291160w^{6}t^{8}-4235343552w^{5}t^{9}-1866396834w^{4}t^{10}+1383790200w^{3}t^{11}+2165221822w^{2}t^{12}+1260866496wt^{13}+405778692t^{14}}{t^{8}(621xyt^{4}+348xw^{4}t+132xw^{3}t^{2}+1626xw^{2}t^{3}+3033xwt^{4}+1056xt^{5}-792y^{2}w^{4}-744y^{2}w^{3}t-1604y^{2}w^{2}t^{2}-1444y^{2}wt^{3}+802y^{2}t^{4}-792yw^{5}+1656yw^{4}t-380yw^{3}t^{2}+5580yw^{2}t^{3}+9762ywt^{4}+1651yt^{5}-288z^{6}-48z^{4}t^{2}-576z^{3}t^{3}+2628z^{2}w^{4}+1296z^{2}w^{3}t+2076z^{2}w^{2}t^{2}+4212z^{2}wt^{3}+836z^{2}t^{4}+36zw^{5}-660zw^{4}t-510zw^{3}t^{2}-2202zw^{2}t^{3}+246zwt^{4}+1554zt^{5}+1584w^{6}+840w^{5}t+34w^{4}t^{2}+4636w^{3}t^{3}+108w^{2}t^{4}-4958wt^{5}-1478t^{6})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.96.3.ba.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 24X^{5}Y^{2}-8X^{6}Z+72X^{4}Y^{2}Z-20X^{5}Z^{2}+48X^{3}Y^{2}Z^{2}-24X^{4}Z^{3}-24X^{2}Y^{2}Z^{3}-16X^{3}Z^{4}-30XY^{2}Z^{4}-6X^{2}Z^{5}-6Y^{2}Z^{5}-XZ^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.96.3.ba.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x^{3}-3x^{2}y-4xy^{2}-2y^{3}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2x^{11}t+24x^{10}yt+124x^{9}y^{2}t+360x^{8}y^{3}t+608x^{7}y^{4}t+448x^{6}y^{5}t-448x^{5}y^{6}t-1664x^{4}y^{7}t-2208x^{3}y^{8}t-1664x^{2}y^{9}t-704xy^{10}t-128y^{11}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle x^{2}y+2xy^{2}+2y^{3}$ |
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.
