Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x z t - y z t - y w t - y t^{2} $ |
| $=$ | $x z w - y z w - y w^{2} - y w t$ |
| $=$ | $x z^{2} - y z^{2} - y z w - y z t$ |
| $=$ | $x y z - y^{2} z - y^{2} w - y^{2} t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 8 x^{6} z + 56 x^{5} y^{2} + 20 x^{5} z^{2} - 168 x^{4} y^{2} z - 24 x^{4} z^{3} + 112 x^{3} y^{2} z^{2} + \cdots + 14 y^{2} z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -14x^{7} + 98x^{5} - 98x^{3} + 14x $ |
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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 |
|---|
| $(0:0:1:0:1)$, $(0:2:-1:1:0)$, $(-2:-2:1:0:0)$, $(-2:0:0:1:0)$ |
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}{2^8\cdot7}\cdot\frac{29812053784895xyt^{12}-1970061681952xw^{13}-18755362634848xw^{12}t-55857051239584xw^{11}t^{2}-17089366205184xw^{10}t^{3}+7930814150392xw^{9}t^{4}-408016610924104xw^{8}t^{5}-1004556593119184xw^{7}t^{6}-654969867524896xw^{6}t^{7}+299823145454546xw^{5}t^{8}+798016561509790xw^{4}t^{9}+466963783796790xw^{3}t^{10}+204863298488520xw^{2}t^{11}+62225589998177xwt^{12}+4397855361291xt^{13}+3944919678336y^{2}w^{12}+11841490440192y^{2}w^{11}t+58395568149632y^{2}w^{10}t^{2}+149452094960896y^{2}w^{9}t^{3}+303186623740448y^{2}w^{8}t^{4}+540823992106496y^{2}w^{7}t^{5}+635481145030592y^{2}w^{6}t^{6}+388141672594816y^{2}w^{5}t^{7}-253812783784504y^{2}w^{4}t^{8}-672379879900288y^{2}w^{3}t^{9}-624745625064792y^{2}w^{2}t^{10}-280769501249776y^{2}wt^{11}-52656880241530y^{2}t^{12}+1972459839168yw^{13}+17773051838752yw^{12}t+52864343037184yw^{11}t^{2}-8237501160224yw^{10}t^{3}-45732065625456yw^{9}t^{4}+431360951768040yw^{8}t^{5}+946981527688896yw^{7}t^{6}+188222518940848yw^{6}t^{7}-886032166956636yw^{5}t^{8}-738687538726914yw^{4}t^{9}+96370379552160yw^{3}t^{10}+160823813136574yw^{2}t^{11}+2560762349850ywt^{12}-17687176732235yt^{13}-1727094849536z^{14}+863547424768z^{12}t^{2}-3454189699072z^{11}t^{3}-281974669312z^{10}t^{4}+986911342592z^{9}t^{5}-2212997627904z^{8}t^{6}-138469703680z^{7}t^{7}+215751983104z^{6}t^{8}-504605179904z^{5}t^{9}+18660655104z^{4}t^{10}-29672079360z^{3}t^{11}-240595028576z^{2}w^{12}-26660114237568z^{2}w^{11}t-78080890868064z^{2}w^{10}t^{2}+52686976211520z^{2}w^{9}t^{3}+157060454311880z^{2}w^{8}t^{4}-514658461280320z^{2}w^{7}t^{5}-1131473745003952z^{2}w^{6}t^{6}-542820612701024z^{2}w^{5}t^{7}+587114732500758z^{2}w^{4}t^{8}+641879991846072z^{2}w^{3}t^{9}+365251938532362z^{2}w^{2}t^{10}+178371292383396z^{2}wt^{11}+39165421782512z^{2}t^{12}+26353376zw^{13}-1979195009120zw^{12}t-6915876933616zw^{11}t^{2}-15535468700752zw^{10}t^{3}-250063285640zw^{9}t^{4}-103634715178680zw^{8}t^{5}-340267926328148zw^{7}t^{6}-227031535037580zw^{6}t^{7}+283438951906838zw^{5}t^{8}+483575568596274zw^{4}t^{9}+102780429728375zw^{3}t^{10}-130375352576875zw^{2}t^{11}-79485853202810zwt^{12}-20128802902230zt^{13}-3940123363904w^{14}-33570601905792w^{13}t-74079672580144w^{12}t^{2}+76056380960000w^{11}t^{3}+58497685868160w^{10}t^{4}-860662440100512w^{9}t^{5}-1134051847978924w^{8}t^{6}+602963391162944w^{7}t^{7}+2064450457443948w^{6}t^{8}+912700804684424w^{5}t^{9}-608558933458861w^{4}t^{10}-586537181524000w^{3}t^{11}-280089082384229w^{2}t^{12}-109203865199262wt^{13}-12772486346010t^{14}}{t^{8}(55223xyt^{4}+29470xw^{5}+99386xw^{4}t+202962xw^{3}t^{2}+112108xw^{2}t^{3}-86799xwt^{4}-79581xt^{5}-57960y^{2}w^{4}-57568y^{2}w^{3}t-155576y^{2}w^{2}t^{2}-145280y^{2}wt^{3}-46250y^{2}t^{4}-28980yw^{5}-83678yw^{4}t-192280yw^{3}t^{2}-51726yw^{2}t^{3}+195082ywt^{4}+116429yt^{5}+25088z^{6}+1792z^{4}t^{2}+21504z^{3}t^{3}+4970z^{2}w^{4}+125608z^{2}w^{3}t+266158z^{2}w^{2}t^{2}+5524z^{2}wt^{3}-133648z^{2}t^{4}-98zw^{5}+28378zw^{4}t+13461zw^{3}t^{2}+78687zw^{2}t^{3}-6706zwt^{4}-57190zt^{5}+58940w^{6}+139832w^{5}t+205437w^{4}t^{2}-174232w^{3}t^{3}-410203w^{2}t^{4}+17994wt^{5}+142454t^{6})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
56.96.3.x.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{7}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 56X^{5}Y^{2}-8X^{6}Z-168X^{4}Y^{2}Z+20X^{5}Z^{2}+112X^{3}Y^{2}Z^{2}-24X^{4}Z^{3}+56X^{2}Y^{2}Z^{3}+16X^{3}Z^{4}-70XY^{2}Z^{4}-6X^{2}Z^{5}+14Y^{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
56.96.3.x.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.
