Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} z - y^{2} z + y z^{2} - z^{3} + z w^{2} $ |
| $=$ | $3 x^{2} y - y^{3} + y^{2} z - y z^{2} + y w^{2}$ |
| $=$ | $3 x^{2} w - y^{2} w + y z w - z^{2} w + w^{3}$ |
| $=$ | $3 x^{3} - x y^{2} + x y z - x z^{2} + x w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} + 2 x^{4} z - 3 x^{3} y^{2} - 8 x^{3} z^{2} - 2 x^{2} y^{3} + 10 x^{2} z^{3} + 2 x y^{3} z + \cdots + z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{3} + x + 1\right) y $ | $=$ | $ -x^{5} + 2x^{4} - 3x^{3} + x^{2} - x $ |
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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 |
|---|
| $(1:-2:-2:1)$, $(1:0:2:1)$, $(1:2:0:1)$, $(0:1:1:1)$, $(0:0:-1:1)$, $(0:-1:0:1)$ |
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 \frac{1}{3^3}\cdot\frac{2782227411454284xyz^{20}+5114524709218401xyz^{19}w+6646303143306000xyz^{18}w^{2}+5650780095770292xyz^{17}w^{3}+3850840550048382xyz^{16}w^{4}+2002621944704886xyz^{15}w^{5}+873427207384782xyz^{14}w^{6}+304155403365744xyz^{13}w^{7}+92434905531549xyz^{12}w^{8}+22970201068602xyz^{11}w^{9}+5332129483455xyz^{10}w^{10}+1003929153642xyz^{9}w^{11}+192293556201xyz^{8}w^{12}+26257382901xyz^{7}w^{13}+4474521081xyz^{6}w^{14}+386619156xyz^{5}w^{15}+71125830xyz^{4}w^{16}+1346571xyz^{3}w^{17}+798174xyz^{2}w^{18}-6324xyzw^{19}-3691xyw^{20}-1815969971746521xz^{21}-4686131223925191xz^{20}w-6366532838760657xz^{19}w^{2}-6137547530572908xz^{18}w^{3}-4303811961836082xz^{17}w^{4}-2422894988567718xz^{16}w^{5}-1063731445794072xz^{15}w^{6}-394251300436590xz^{14}w^{7}-116539876731384xz^{13}w^{8}-31106766134610xz^{12}w^{9}-6614294900454xz^{11}w^{10}-1445429877300xz^{10}w^{11}-218711919393xz^{9}w^{12}-44239196196xz^{8}w^{13}-3476071098xz^{7}w^{14}-1057617567xz^{6}w^{15}+17683164xz^{5}w^{16}-21192759xz^{4}w^{17}+2129202xz^{3}w^{18}-390429xz^{2}w^{19}+32336xzw^{20}-4096xw^{21}+815549908109994y^{2}z^{20}+810864229502106y^{2}z^{19}w+912289944173922y^{2}z^{18}w^{2}+388829503483416y^{2}z^{17}w^{3}+191201173246242y^{2}z^{16}w^{4}-8960465785914y^{2}z^{15}w^{5}-13499917103574y^{2}z^{14}w^{6}-20705296721256y^{2}z^{13}w^{7}-5496608226420y^{2}z^{12}w^{8}-2876496321060y^{2}z^{11}w^{9}-379757661876y^{2}z^{10}w^{10}-199498583556y^{2}z^{9}w^{11}-6254306784y^{2}z^{8}w^{12}-9887479083y^{2}z^{7}w^{13}+708765957y^{2}z^{6}w^{14}-404926128y^{2}z^{5}w^{15}+54493020y^{2}z^{4}w^{16}-12658140y^{2}z^{3}w^{17}+1834173y^{2}z^{2}w^{18}-267022y^{2}zw^{19}+20354y^{2}w^{20}-815549908109994yz^{21}-2060359179830406yz^{20}w-3575479250454453yz^{19}w^{2}-3830470695859518yz^{18}w^{3}-3272265587176182yz^{17}w^{4}-2066235441485310yz^{16}w^{5}-1098286332137982yz^{15}w^{6}-452224608022854yz^{14}w^{7}-163716017140800yz^{13}w^{8}-46102434530157yz^{12}w^{9}-12438020776698yz^{11}w^{10}-2486215778355yz^{10}w^{11}-577641964194yz^{9}w^{12}-75127132989yz^{8}w^{13}-18679892031yz^{7}w^{14}-766470249yz^{6}w^{15}-515901150yz^{5}w^{16}+33144012yz^{4}w^{17}-12672504yz^{3}w^{18}+1584958yz^{2}w^{19}-257794yzw^{20}+20327yw^{21}+815549908109994z^{22}+2343596690509992z^{21}w+3363625347111792z^{20}w^{2}+3310253944855659z^{19}w^{3}+2367447434238186z^{18}w^{4}+1307405425325832z^{17}w^{5}+563656346789940z^{16}w^{6}+192089379884292z^{15}w^{7}+52444536314328z^{14}w^{8}+11239181841618z^{13}w^{9}+2080605780072z^{12}w^{10}+262743126138z^{11}w^{11}+38352590208z^{10}w^{12}-4096453014z^{9}w^{13}+164161728z^{8}w^{14}-504791568z^{7}w^{15}+40532130z^{6}w^{16}-25002405z^{5}w^{17}+2905893z^{4}w^{18}-660250z^{3}w^{19}+93732z^{2}w^{20}-8318zw^{21}}{w^{6}z^{3}(248029128xyz^{11}+283668804xyz^{10}w+277569909xyz^{9}w^{2}+149408685xyz^{8}w^{3}+70563453xyz^{7}w^{4}+21019566xyz^{6}w^{5}+5713272xyz^{5}w^{6}+916533xyz^{4}w^{7}+146205xyz^{3}w^{8}+10215xyz^{2}w^{9}+795xyzw^{10}+15xyw^{11}-161889516xz^{12}-305310465xz^{11}w-278542557xz^{10}w^{2}-191115675xz^{9}w^{3}-81428322xz^{8}w^{4}-30512241xz^{7}w^{5}-6548400xz^{6}w^{6}-1557081xz^{5}w^{7}-115425xz^{4}w^{8}-27684xz^{3}w^{9}+930xz^{2}w^{10}-144xzw^{11}+72704385y^{2}z^{11}+21786732y^{2}z^{10}w+31636278y^{2}z^{9}w^{2}-5496048y^{2}z^{8}w^{3}+1583271y^{2}z^{7}w^{4}-3317094y^{2}z^{6}w^{5}-37953y^{2}z^{5}w^{6}-334857y^{2}z^{4}w^{7}+20268y^{2}z^{3}w^{8}-11577y^{2}z^{2}w^{9}+1017y^{2}zw^{10}-90y^{2}w^{11}-72704385yz^{12}-133176312yz^{11}w-191683044yz^{10}w^{2}-137202975yz^{9}w^{3}-89064096yz^{8}w^{4}-32949825yz^{7}w^{5}-12676505yz^{6}w^{6}-2282388yz^{5}w^{7}-648024yz^{4}w^{8}-23793yz^{3}w^{9}-13791yz^{2}w^{10}+861yzw^{11}-91yw^{12}+72704385z^{13}+158426280z^{12}w+155258316z^{11}w^{2}+104124870z^{10}w^{3}+45972687z^{9}w^{4}+14220216z^{8}w^{5}+2999863z^{7}w^{6}+199683z^{6}w^{7}+8949z^{5}w^{8}-25374z^{4}w^{9}+780z^{3}w^{10}-657z^{2}w^{11}+38zw^{12})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
18.108.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{5}-3X^{3}Y^{2}-2X^{2}Y^{3}+2X^{4}Z+2XY^{3}Z-8X^{3}Z^{2}+9XY^{2}Z^{2}-2Y^{3}Z^{2}+10X^{2}Z^{3}-3Y^{2}Z^{3}-7XZ^{4}+Z^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
18.108.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -y^{3}z+y^{2}z^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{74}{729}y^{12}+\frac{16}{243}y^{11}z+\frac{134}{243}y^{11}w+\frac{661}{243}y^{10}z^{2}-\frac{212}{243}y^{10}zw+\frac{482}{243}y^{10}w^{2}-\frac{7577}{729}y^{9}z^{3}-\frac{7}{243}y^{9}z^{2}w-\frac{1060}{243}y^{9}zw^{2}+\frac{1526}{729}y^{9}w^{3}+\frac{1367}{81}y^{8}z^{4}-\frac{302}{81}y^{8}z^{3}w-\frac{58}{27}y^{8}z^{2}w^{2}-\frac{148}{27}y^{8}zw^{3}+\frac{236}{243}y^{8}w^{4}-\frac{359}{27}y^{7}z^{5}+\frac{334}{27}y^{7}z^{4}w+\frac{908}{81}y^{7}z^{3}w^{2}+\frac{230}{81}y^{7}z^{2}w^{3}-\frac{644}{243}y^{7}zw^{4}+\frac{56}{243}y^{7}w^{5}+\frac{665}{243}y^{6}z^{6}-\frac{1057}{81}y^{6}z^{5}w-\frac{508}{81}y^{6}z^{4}w^{2}+\frac{962}{243}y^{6}z^{3}w^{3}+\frac{644}{243}y^{6}z^{2}w^{4}-\frac{160}{243}y^{6}zw^{5}+\frac{16}{729}y^{6}w^{6}+\frac{32}{9}y^{5}z^{7}+\frac{47}{9}y^{5}z^{6}w-\frac{110}{27}y^{5}z^{5}w^{2}-\frac{470}{81}y^{5}z^{4}w^{3}-\frac{152}{243}y^{5}z^{3}w^{4}+\frac{80}{81}y^{5}z^{2}w^{5}-\frac{16}{243}y^{5}zw^{6}-\frac{241}{81}y^{4}z^{8}+\frac{11}{81}y^{4}z^{7}w+\frac{452}{81}y^{4}z^{6}w^{2}+\frac{8}{3}y^{4}z^{5}w^{3}-\frac{364}{243}y^{4}z^{4}w^{4}-\frac{200}{243}y^{4}z^{3}w^{5}+\frac{32}{243}y^{4}z^{2}w^{6}+\frac{748}{729}y^{3}z^{9}-\frac{82}{81}y^{3}z^{8}w-\frac{190}{81}y^{3}z^{7}w^{2}+\frac{68}{243}y^{3}z^{6}w^{3}+\frac{388}{243}y^{3}z^{5}w^{4}+\frac{64}{243}y^{3}z^{4}w^{5}-\frac{112}{729}y^{3}z^{3}w^{6}-\frac{62}{243}y^{2}z^{10}+\frac{122}{243}y^{2}z^{9}w+\frac{14}{27}y^{2}z^{8}w^{2}-\frac{62}{81}y^{2}z^{7}w^{3}-\frac{136}{243}y^{2}z^{6}w^{4}+\frac{8}{81}y^{2}z^{5}w^{5}+\frac{32}{243}y^{2}z^{4}w^{6}+\frac{10}{243}yz^{11}-\frac{26}{243}yz^{10}w-\frac{22}{243}yz^{9}w^{2}+\frac{22}{81}yz^{8}w^{3}+\frac{28}{243}yz^{7}w^{4}-\frac{40}{243}yz^{6}w^{5}-\frac{16}{243}yz^{5}w^{6}-\frac{2}{729}z^{12}+\frac{2}{243}z^{11}w+\frac{2}{243}z^{10}w^{2}-\frac{22}{729}z^{9}w^{3}-\frac{4}{243}z^{8}w^{4}+\frac{8}{243}z^{7}w^{5}+\frac{16}{729}z^{6}w^{6}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{5}{9}y^{4}+\frac{1}{9}y^{3}z-\frac{7}{9}y^{3}w+y^{2}z^{2}+\frac{2}{3}y^{2}zw-\frac{2}{9}y^{2}w^{2}-\frac{5}{9}yz^{3}+\frac{1}{3}yz^{2}w+\frac{2}{9}yzw^{2}+\frac{1}{9}z^{4}-\frac{1}{9}z^{3}w-\frac{2}{9}z^{2}w^{2}$ |
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.
