Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y z - x z^{2} - x z w - y^{2} z + y z^{2} - z^{3} + z w^{2} $ |
| $=$ | $x y^{2} - x y z - x y w - y^{3} + y^{2} z - y z^{2} + y w^{2}$ |
| $=$ | $x y w - x z w - x w^{2} - y^{2} w + y z w - z^{2} w + w^{3}$ |
| $=$ | $x^{2} y - x^{2} z - x^{2} w - x y^{2} + x y z - x z^{2} + x w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{3} y - x^{3} z - 2 x^{2} y^{2} + 4 x^{2} y z - 2 x^{2} z^{2} + 2 x y^{3} - 3 x y^{2} z + \cdots - y^{2} z^{2} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{3} + x + 1\right) y $ | $=$ | $ x^{5} + 2x^{4} + 2x^{3} + x^{2} $ |
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This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(1:1:0:0)$, $(1:0:0:0)$, $(-1:0:1:0)$, $(1:0:0:1)$, $(1:0:-1:1)$, $(1:1:0:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 108 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2\,\frac{524288x^{22}-13369344x^{21}w+162398208x^{20}w^{2}-1248067584x^{19}w^{3}+6797721600x^{18}w^{4}-27854143488x^{17}w^{5}+88955633664x^{16}w^{6}-226326085632x^{15}w^{7}+464931145728x^{14}w^{8}-776783047680x^{13}w^{9}+1058044626432x^{12}w^{10}-1172523368448x^{11}w^{11}+1050374939904x^{10}w^{12}-751997748096x^{9}w^{13}+422830290240x^{8}w^{14}-182040225984x^{7}w^{15}+57839562432x^{6}w^{16}-12832557840x^{5}w^{17}+1815916752x^{4}w^{18}-138313980x^{3}w^{19}+3957903x^{2}w^{20}-191102976xz^{20}w-764411904xz^{19}w^{2}-47775744xz^{18}w^{3}+4825350144xz^{17}w^{4}+10809262080xz^{16}w^{5}+9029615616xz^{15}w^{6}-964472832xz^{14}w^{7}-8192047104xz^{13}w^{8}-6631870464xz^{12}w^{9}-1492535808xz^{11}w^{10}+1334444544xz^{10}w^{11}+1322002944xz^{9}w^{12}+437439744xz^{8}w^{13}-39788928xz^{7}w^{14}-89098272xz^{6}w^{15}-37439280xz^{5}w^{16}-4240296xz^{4}w^{17}+1441908xz^{3}w^{18}+838566xz^{2}w^{19}+1236762xzw^{20}+613926xw^{21}-573308928yz^{19}w^{2}-3009871872yz^{18}w^{3}-5709201408yz^{17}w^{4}-3069591552yz^{16}w^{5}+4843266048yz^{15}w^{6}+8859414528yz^{14}w^{7}+4793997312yz^{13}w^{8}-863695872yz^{12}w^{9}-2601828864yz^{11}w^{10}-1460851200yz^{10}w^{11}-159252480yz^{9}w^{12}+259887744yz^{8}w^{13}+151488576yz^{7}w^{14}+29461536yz^{6}w^{15}-5782320yz^{5}w^{16}-5846040yz^{4}w^{17}+295596yz^{3}w^{18}+189054yz^{2}w^{19}-288603yzw^{20}-191102976z^{21}w-573308928z^{20}w^{2}+1051066368z^{19}w^{3}+6688604160z^{18}w^{4}+9519316992z^{17}w^{5}+41803776z^{16}w^{6}-13616087040z^{15}w^{7}-13789274112z^{14}w^{8}-2026736640z^{13}w^{9}+5898147840z^{12}w^{10}+4935043584z^{11}w^{11}+1160428032z^{10}w^{12}-763713792z^{9}w^{13}-699198912z^{8}w^{14}-178318368z^{7}w^{15}+28012176z^{6}w^{16}+38673720z^{5}w^{17}+11160180z^{4}w^{18}-898938z^{3}w^{19}-79461z^{2}w^{20}-948159zw^{21}-402489w^{22}}{w^{7}(4x^{4}w^{11}-30x^{3}w^{12}+87x^{2}w^{13}-221184xz^{14}-1105920xz^{13}w-1935360xz^{12}w^{2}-850944xz^{11}w^{3}+1526784xz^{10}w^{4}+2215680xz^{9}w^{5}+949888xz^{8}w^{6}-51200xz^{7}w^{7}-137152xz^{6}w^{8}-23600xz^{5}w^{9}+1048xz^{4}w^{10}-4xz^{3}w^{11}+16xz^{2}w^{12}+28xzw^{13}-100xw^{14}+221184yz^{14}+552960yz^{13}w-221184yz^{12}w^{2}-1929216yz^{11}w^{3}-2070528yz^{10}w^{4}-446976yz^{9}w^{5}+515968yz^{8}w^{6}+298688yz^{7}w^{7}+20608yz^{6}w^{8}-9280yz^{5}w^{9}+40yz^{4}w^{10}-28yz^{3}w^{11}+18yz^{2}w^{12}-21yzw^{13}-221184z^{15}-995328z^{14}w-1105920z^{13}w^{2}+1305600z^{12}w^{3}+3743232z^{11}w^{4}+2204928z^{10}w^{5}-926336z^{9}w^{6}-1440320z^{8}w^{7}-358496z^{7}w^{8}+91664z^{6}w^{9}+34040z^{5}w^{10}-1140z^{4}w^{11}+48z^{3}w^{12}-27z^{2}w^{13}-7zw^{14}+39w^{15})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
$X_{\pm1}(18)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{3}Y-2X^{2}Y^{2}+2XY^{3}-X^{3}Z+4X^{2}YZ-3XY^{2}Z+Y^{3}Z-2X^{2}Z^{2}+2XYZ^{2}-Y^{2}Z^{2}-XZ^{3} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
$X_{\pm1}(18)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y^{2}z+y^{2}w-2yz^{2}-4yzw-2yw^{2}+z^{2}w+2zw^{2}+w^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y-z-w$ |
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.
