Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} t - x z t - y z t $ |
| $=$ | $x^{2} w - x z w - y z w$ |
| $=$ | $x^{2} z - x z^{2} - y z^{2}$ |
| $=$ | $x^{3} - x^{2} z - x y z$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} y - 4 x^{4} y z + 11 x^{3} y z^{2} - 11 x^{2} y^{2} z^{2} - 10 x^{2} y z^{3} - x^{2} z^{4} + \cdots - 2 z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + x^{2} + 1\right) y $ | $=$ | $ 2x^{6} - 2x^{5} + 11x^{4} - 10x^{3} + 20x^{2} - 11x + 8 $ |
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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 |
|---|
| $(0:1:0:0:0)$, $(0:0:0:-3/2:1)$, $(0:0:0:0:1)$, $(0:0:0:1/3:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 11\,\frac{17519694xzt^{6}+22031343xw^{3}t^{4}-17200407xw^{2}t^{5}-43249388xwt^{6}-24984278xt^{7}-5832y^{8}-138510y^{7}w-600939y^{7}t-764721y^{6}w^{2}-3399327y^{6}wt+453519y^{6}t^{2}-1974861y^{5}w^{3}-16035327y^{5}w^{2}t+585306y^{5}wt^{2}+5726214y^{5}t^{3}-3642084y^{4}w^{4}-32360796y^{4}w^{3}t-13436442y^{4}w^{2}t^{2}+13451832y^{4}wt^{3}+21677454y^{4}t^{4}-56256444y^{3}w^{4}t-47728197y^{3}w^{3}t^{2}+42575598y^{3}w^{2}t^{3}+93574998y^{3}wt^{4}+7003647y^{3}t^{5}-152478531y^{2}w^{4}t^{2}-75744369y^{2}w^{3}t^{3}+337734234y^{2}w^{2}t^{4}+52068699y^{2}wt^{5}-21053090y^{2}t^{6}-203347557yw^{4}t^{3}+294648723yw^{3}t^{4}+311270556yw^{2}t^{5}-52997296ywt^{6}-25323630yt^{7}+57119562z^{2}wt^{5}+8415933z^{2}t^{6}+3087315zw^{7}+8812881zw^{6}t+10674666zw^{5}t^{2}+35921691zw^{4}t^{3}+10334358zw^{3}t^{4}+10787898zw^{2}t^{5}-32687778zwt^{6}+1163970zt^{7}-4998753w^{8}-27700056w^{7}t-103268682w^{6}t^{2}-143070570w^{5}t^{3}+90535239w^{4}t^{4}+245492910w^{3}t^{5}+27602219w^{2}t^{6}-35626821wt^{7}-658449t^{8}}{10896051xzt^{6}+41676624xw^{3}t^{4}+131527389xw^{2}t^{5}+14304323xwt^{6}+1961841xt^{7}-90639y^{6}t^{2}-424521y^{5}wt^{2}+2330451y^{5}t^{3}+23328y^{4}w^{3}t-2143503y^{4}w^{2}t^{2}+8350614y^{4}wt^{3}-15468039y^{4}t^{4}+75087y^{3}w^{4}t-4444551y^{3}w^{3}t^{2}+32675967y^{3}w^{2}t^{3}-56858949y^{3}wt^{4}+3113403y^{3}t^{5}-9867501y^{2}w^{4}t^{2}+32944671y^{2}w^{3}t^{3}-203069025y^{2}w^{2}t^{4}-17325210y^{2}wt^{5}+16331796y^{2}t^{6}+46184148yw^{4}t^{3}-220205700yw^{3}t^{4}-96575007yw^{2}t^{5}+63490459ywt^{6}-2847338yt^{7}-52082616z^{2}wt^{5}-14427537z^{2}t^{6}-54675zw^{7}+798012zw^{6}t+14441004zw^{5}t^{2}+16008759zw^{4}t^{3}+37943550zw^{3}t^{4}-78711786zw^{2}t^{5}+69610041zwt^{6}+69984w^{8}-1618380w^{7}t-9492876w^{6}t^{2}-69594849w^{5}t^{3}-203237235w^{4}t^{4}-102446919w^{3}t^{5}+95652735w^{2}t^{6}-12073500wt^{7}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
$X_0(33)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{5}Y-4X^{4}YZ+11X^{3}YZ^{2}-11X^{2}Y^{2}Z^{2}-10X^{2}YZ^{3}-X^{2}Z^{4}+13XYZ^{4}+XZ^{5}-2Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
$X_0(33)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x+z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2x^{2}z^{2}-2xz^{3}-\frac{11}{3}xz^{2}t+5z^{4}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.
