Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x v + y^{2} + y v $ |
| $=$ | $y^{2} - z^{2} - z t + z u$ |
| $=$ | $x z + x t - y w$ |
| $=$ | $y z + y t + z v + w v + t v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{6} z^{2} - 4 x^{5} y z^{2} - 9 x^{4} y^{4} - x^{4} y^{2} z^{2} - x^{4} z^{4} - 36 x^{3} y^{5} + \cdots + 4 y^{6} z^{2} $ |
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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.
| Canonical model |
|---|
| $(0:0:1/3:0:2/3:1:0)$, $(0:0:0:-2:1:0:0)$, $(0:0:0:0:1:0:0)$, $(0:0:1/3:-2:2/3:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2}{3^4}\cdot\frac{5282287344xt^{10}v-43983631800xt^{9}uv-56459384283114xt^{8}v^{3}+199064593900782xt^{7}uv^{3}+23183160296061492xt^{6}v^{5}-35253185818510386xt^{5}uv^{5}-383244585653474982xt^{4}v^{7}+216888375894797772xt^{3}uv^{7}-10148855777324341452xt^{2}v^{9}+5187239068896665286xtuv^{9}+47041818504xu^{10}v-440399988250xu^{8}v^{3}-2247717047461488xu^{6}v^{5}+161495510819657912xu^{4}v^{7}+5015650698340363898xu^{2}v^{9}+81234937478668720578xv^{11}+202963040208yu^{10}v-3920745887250yu^{8}v^{3}+138849108427794yu^{6}v^{5}+19399855153702110yu^{4}v^{7}+11009086409683733286yu^{2}v^{9}+115392346256925030288yv^{11}+1340884692zt^{11}+2178451533132zt^{9}v^{2}-3463395313455018zt^{7}v^{4}+117246011547522852zt^{5}v^{6}+1818660797194684680zt^{3}v^{8}+103674783648319352532ztv^{10}+19382340732zu^{11}-2924862275020zu^{9}v^{2}+2307449140230414zu^{7}v^{4}-159530572968321190zu^{5}v^{6}-5767058715404572234zu^{3}v^{8}-61619678276058401340zuv^{10}-461290788wt^{11}-7588741284wt^{9}v^{2}+56441936327130wt^{7}v^{4}-23222417272650576wt^{5}v^{6}+430171878949393716wt^{3}v^{8}+9470966663634544296wtv^{10}+5965764wu^{11}+157334905820wu^{9}v^{2}-16518260795232wu^{7}v^{4}+7884130905226610wu^{5}v^{6}-185337215838304984wu^{3}v^{8}-691246226379445032wuv^{10}+59049t^{12}-1396154556t^{11}u-1814103378t^{10}v^{2}+13077807124788t^{9}uv^{2}+10922393263761t^{8}v^{4}-11586504590499456t^{7}uv^{4}-8326937311014996t^{6}v^{6}+342267399889677234t^{5}uv^{6}+300389908339526940t^{4}v^{8}+4853898665837297658t^{3}uv^{8}+2263091432542343892t^{2}v^{10}-11065721736tu^{11}+1735936298768tu^{9}v^{2}-33877629251784tu^{7}v^{4}+6124165823864390tu^{5}v^{6}+444752641187235476tu^{3}v^{8}+15304475117765065086tuv^{10}+928606389u^{12}-345497347198u^{10}v^{2}-12172021821619u^{8}v^{4}+9989516253868874u^{6}v^{6}-383535918067398404u^{4}v^{8}-11642931759901272472u^{2}v^{10}-11249394499984078113v^{12}}{v(118098xt^{10}-1220346xt^{9}u-207170136xt^{8}v^{2}+138209652xt^{7}uv^{2}-3783084264xt^{6}v^{4}+966237984xt^{5}uv^{4}-7048102653xt^{4}v^{6}-19869184062xt^{3}uv^{6}+356737015899xt^{2}v^{8}+323873515323xtuv^{8}+11260932xu^{8}v^{2}+852171882xu^{6}v^{4}+17177528080xu^{4}v^{6}-266375477879xu^{2}v^{8}-2721263237145xv^{10}-98307yu^{8}v^{2}+68057775yu^{6}v^{4}-1254426606yu^{4}v^{6}+69895421130yu^{2}v^{8}-3742090205346yv^{10}+39680928zt^{9}v+2040982758zt^{7}v^{3}+24639782283zt^{5}v^{5}-48846256848zt^{3}v^{7}-3428332586469ztv^{9}-11293701zu^{9}v-885383835zu^{7}v^{3}-19233581123zu^{5}v^{5}+182292379987zu^{3}v^{7}+2278647981117zuv^{9}-118098wt^{9}v+206540280wt^{7}v^{3}+3255010515wt^{5}v^{5}-3483055890wt^{3}v^{7}-402987204657wtv^{9}+32769wu^{9}v-22928862wu^{7}v^{3}-1400609543wu^{5}v^{5}-24885920657wu^{3}v^{7}+128420944389wuv^{9}+129396042t^{9}uv+90187506t^{8}v^{3}+3824105094t^{7}uv^{3}+1139224338t^{6}v^{5}+26738742897t^{5}uv^{5}-11724324264t^{4}v^{7}-342557953497t^{3}uv^{7}-368033952969t^{2}v^{9}+32769tu^{9}v-11799006tu^{7}v^{3}-875500703tu^{5}v^{5}-61910732tu^{3}v^{7}+67122035694tuv^{9}+32769u^{10}v-34222563u^{8}v^{3}-2105831444u^{6}v^{5}-29657419798u^{4}v^{7}+311957102305u^{2}v^{9}+339542945991v^{11})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.7.fm.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 3z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -X^{6}Z^{2}-4X^{5}YZ^{2}-9X^{4}Y^{4}-X^{4}Y^{2}Z^{2}-X^{4}Z^{4}-36X^{3}Y^{5}+16X^{3}Y^{3}Z^{2}-4X^{3}YZ^{4}-45X^{2}Y^{6}+25X^{2}Y^{4}Z^{2}-5X^{2}Y^{2}Z^{4}-18XY^{7}+14XY^{5}Z^{2}-2XY^{3}Z^{4}+4Y^{6}Z^{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.
