Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ - z v + t u $ |
| $=$ | $x u + y v$ |
| $=$ | $x u - x v + w u$ |
| $=$ | $x z + y t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{7} + 9 x^{6} z + 18 x^{5} y^{2} + 24 x^{4} y^{2} z + 5 x^{3} y^{4} + 9 x^{3} y^{2} z^{2} + \cdots + 5 y^{4} z^{3} $ |
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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 |
|---|
| $(0:0:1:0:0:0:0)$, $(0:0:0:0:0:1:0)$, $(0:0:0:0:1:0:0)$, $(0:0:0:0:0:0:1)$ |
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{390625z^{12}-2343750z^{11}t-58359375z^{11}u+285390625z^{11}v-15150703125z^{10}tv-20715281250z^{10}uv+306006031250z^{10}v^{2}-1795912031250z^{9}tv^{2}-2853320006250z^{9}uv^{2}+9023123243750z^{9}v^{3}-52163416978125z^{8}tv^{3}-53256678583125z^{8}uv^{3}+108214265641875z^{8}v^{4}-511006280925000z^{7}tv^{4}-442535490313125z^{7}uv^{4}+620975263362875z^{7}v^{5}-2729546489920125z^{6}tv^{5}-2099241346695900z^{6}uv^{5}+2172429999631450z^{6}v^{6}-9325715158736625z^{5}tv^{6}-6540946621696200z^{5}uv^{6}+5133656364396700z^{5}v^{7}-22328315926702800z^{4}tv^{7}-14502684546752560z^{4}uv^{7}+8697730454852125z^{4}v^{8}-39635819843713285z^{3}tv^{8}-24067013369267729z^{3}uv^{8}+10943883900131753z^{3}v^{9}-53796278746084620z^{2}tv^{9}-29865271884765625z^{2}uv^{9}+12175372333857271z^{2}v^{10}+1708593750zt^{11}+3018515625zt^{10}v+2631234375zt^{9}v^{2}+6526828125zt^{8}v^{3}+6647568750zt^{7}v^{4}-111536250zt^{6}v^{5}+1903831225zt^{5}v^{6}+3133092592150zt^{4}v^{7}+213039919826825zt^{3}v^{8}+4065469521103063zt^{2}v^{9}-54396354744140625ztv^{10}-27055178802734375zuv^{10}-1586128281250000zv^{11}+18893532304687500w^{2}v^{10}+284765625t^{12}+1537734375t^{11}v+2836265625t^{10}v^{2}+4159856250t^{9}v^{3}+4391769375t^{8}v^{4}+4699042875t^{7}v^{5}+5600285450t^{6}v^{6}+76905981800t^{5}v^{7}+15207730459620t^{4}v^{8}+536353170048979t^{3}v^{9}+7077580595703125t^{2}v^{10}-7756488515625000tv^{11}+284765625u^{12}-1708593750u^{11}v+7688671875u^{10}v^{2}-38158593750u^{9}v^{3}+185382421875u^{8}v^{4}-857714062500u^{7}v^{5}+3884203125000u^{6}v^{6}-17362729687500u^{5}v^{7}+76718422265625u^{4}v^{8}-335782525781250u^{3}v^{9}+1458644424609375u^{2}v^{10}-6297844099218750uv^{11}+390625v^{12}}{78125z^{11}u-390625z^{11}v-312500z^{10}tv-1109375z^{10}uv-1265625z^{10}v^{2}+7359375z^{9}tv^{2}+959375z^{9}uv^{2}+5750000z^{9}v^{3}+8396875z^{8}tv^{3}+3730625z^{8}uv^{3}+12096250z^{8}v^{4}-2914375z^{7}tv^{4}+2103250z^{7}uv^{4}+8346875z^{7}v^{5}-7430375z^{6}tv^{5}+829075z^{6}uv^{5}+2961225z^{6}v^{6}-7319925z^{5}tv^{6}+39770z^{5}uv^{6}+442955z^{5}v^{7}-3082610z^{4}tv^{7}+86115z^{4}uv^{7}+244362z^{4}v^{8}-666689z^{3}tv^{8}-64255z^{3}uv^{8}-106650z^{3}v^{9}-53753z^{2}tv^{9}-64255z^{2}v^{10}-6834375zt^{8}v^{3}-8656875zt^{7}v^{4}+1135025zt^{5}v^{6}+303595zt^{4}v^{7}-247374zt^{3}v^{8}-192765zt^{2}v^{9}-2278125t^{9}v^{3}-4100625t^{8}v^{4}-5740875t^{7}v^{5}-2606075t^{6}v^{6}-453555t^{5}v^{7}+53753t^{4}v^{8}+64255t^{3}v^{9}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.7.mb.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 9X^{7}+9X^{6}Z+18X^{5}Y^{2}+24X^{4}Y^{2}Z+5X^{3}Y^{4}+9X^{3}Y^{2}Z^{2}+15X^{2}Y^{4}Z-3X^{2}Y^{2}Z^{3}+15XY^{4}Z^{2}-3XY^{2}Z^{4}+5Y^{4}Z^{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.
