Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x z - x w + y t $ |
| $=$ | $18 x y - 2 z^{2} - 2 z w - t^{2}$ |
| $=$ | $24 x^{2} - 6 y^{2} - z t + w t$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{7} - 4 x^{5} y^{2} + 6 x^{4} y z^{2} - 4 x^{3} y^{4} - 9 x y^{2} z^{4} + 12 y^{5} z^{2} $ |
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This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps to other modular curves
$j$-invariant map
of degree 72 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^3\cdot3^3\,\frac{167392xw^{8}t+2155528xw^{6}t^{3}+3533868xw^{4}t^{5}+1226460xw^{2}t^{7}+118773xt^{9}-360yz^{2}w^{7}-212760yz^{2}w^{5}t^{2}-831276yz^{2}w^{3}t^{4}-473508yz^{2}wt^{6}-24yzw^{8}-294848yzw^{6}t^{2}-1941048yzw^{4}t^{4}-2214924yzw^{2}t^{6}-319041yzt^{8}+384yw^{9}-164680yw^{7}t^{2}-2012616yw^{5}t^{4}-2965476yw^{3}t^{6}-585720ywt^{8}}{t(128xw^{8}+9728xw^{6}t^{2}+53784xw^{4}t^{4}+37356xw^{2}t^{6}+6129xt^{8}-432yz^{2}w^{5}t-7452yz^{2}w^{3}t^{3}-13707yz^{2}wt^{5}-496yzw^{6}t-12348yzw^{4}t^{3}-41601yzw^{2}t^{5}-16461yzt^{7}-128yw^{7}t-9360yw^{5}t^{3}-49020yw^{3}t^{5}-26235ywt^{7})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
48.72.5.o.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -X^{7}-4X^{5}Y^{2}+6X^{4}YZ^{2}-4X^{3}Y^{4}-9XY^{2}Z^{4}+12Y^{5}Z^{2} $ |
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.
