Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x z w + x z t - y z t $ |
| $=$ | $x w^{2} + x w t - y w t$ |
| $=$ | $x w t + x t^{2} - y t^{2}$ |
| $=$ | $x^{2} w + x^{2} t - x y t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} y^{2} + 9 x^{4} y z + 6 x^{2} y^{2} z^{2} + 6 x^{2} y z^{3} - 3 x^{2} z^{4} + 5 y^{2} z^{4} + 5 y z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 6x^{8} + 30x^{6} + 49x^{4} + 54x^{2} + 20 $ |
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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.
| Embedded model |
|---|
| $(0:0:0:-1:1)$, $(1:0:0:0:0)$, $(-1:-1:1:0:0)$, $(0:0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{455625x^{11}+2328750x^{9}t^{2}+2986875x^{7}t^{4}-2193750x^{5}t^{6}-4428750x^{3}t^{8}+4352500xt^{10}+2885625yz^{10}+10985625yz^{8}t^{2}-118125yz^{6}t^{4}+4704975yz^{4}t^{6}-8481270yz^{2}t^{8}+3388416yw^{10}+8448000yw^{9}t+42593280yw^{8}t^{2}+111413504yw^{7}t^{3}+153107520yw^{6}t^{4}+119963520yw^{5}t^{5}+3463456yw^{4}t^{6}-68918160yw^{3}t^{7}-74838600yw^{2}t^{8}-17568784ywt^{9}-7784694yt^{10}-1518750z^{11}+4961250z^{9}t^{2}-3543750z^{7}t^{4}+8749350z^{5}t^{6}-2120220z^{3}t^{8}+9014272zw^{10}+30976000zw^{9}t+77054976zw^{8}t^{2}+134897152zw^{7}t^{3}+117406272zw^{6}t^{4}+17890176zw^{5}t^{5}-117043712zw^{4}t^{6}-152941440zw^{3}t^{7}-98280488zw^{2}t^{8}-20935920zwt^{9}-27804zt^{10}}{6750yz^{6}t^{4}-4275yz^{4}t^{6}-90yz^{2}t^{8}-1792yw^{10}-3200yw^{9}t+7168yw^{8}t^{2}+22432yw^{7}t^{3}+12672yw^{6}t^{4}-17240yw^{5}t^{5}-27488yw^{4}t^{6}-11930yw^{3}t^{7}+1555yw^{2}t^{8}+2594ywt^{9}+541yt^{10}+6750z^{7}t^{4}-6525z^{5}t^{6}-915z^{3}t^{8}-4864zw^{10}-14400zw^{9}t+64zw^{8}t^{2}+49936zw^{7}t^{3}+79168zw^{6}t^{4}+51940zw^{5}t^{5}+13620zw^{4}t^{6}+1799zw^{3}t^{7}+2209zw^{2}t^{8}+668zwt^{9}-254zt^{10}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.72.3.ca.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 9X^{4}Y^{2}+9X^{4}YZ+6X^{2}Y^{2}Z^{2}+6X^{2}YZ^{3}-3X^{2}Z^{4}+5Y^{2}Z^{4}+5YZ^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.72.3.ca.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -t^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -9x^{4}wt^{3}-5x^{4}t^{4}-6x^{2}wt^{5}-3x^{2}t^{6}-5wt^{7}-3t^{8}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle xt$ |
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.
