Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 2 x z w - x w t - y w^{2} + y w t $ |
| $=$ | $2 x z^{2} - x z t - y z w + y z t$ |
| $=$ | $2 x z t - x t^{2} - y w t + y t^{2}$ |
| $=$ | $x^{2} w + x^{2} t + 2 x y z + x y t - w^{2} t + w t^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} z + 3 x^{3} y^{2} + x^{3} z^{2} - 18 x^{2} y^{2} z + 2 x^{2} z^{3} + 30 x y^{2} z^{2} + \cdots - 12 y^{2} z^{3} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -3x^{7} + 21x^{5} - 21x^{3} + 3x $ |
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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 |
|---|
| $(0:0:1/2:1:1)$, $(0:1:0:0:0)$, $(1:1:0:0:0)$, $(1:0:0:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2187x^{14}-4374x^{12}t^{2}+17010x^{10}t^{4}-57672x^{8}t^{6}+217458x^{6}t^{8}-825696x^{4}t^{10}+3144132x^{2}t^{12}-4374xy^{13}+33534xy^{11}t^{2}-25272xy^{9}t^{4}+486648xy^{7}t^{6}-214974xy^{5}t^{8}+4586310xy^{3}t^{10}-1996896xyt^{12}-14580y^{12}t^{2}-42768y^{10}t^{4}-237816y^{8}t^{6}-450144y^{6}t^{8}-2217132y^{4}t^{10}-3946848y^{2}t^{12}+640zw^{13}+128zw^{12}t+4352zw^{11}t^{2}+256zw^{10}t^{3}+43904zw^{9}t^{4}+4480zw^{8}t^{5}+232960zw^{7}t^{6}+3592zw^{6}t^{7}+1216704zw^{5}t^{8}+12128zw^{4}t^{9}+4893120zw^{3}t^{10}-214360zw^{2}t^{11}+13187584zwt^{12}+640zt^{13}-224w^{14}-64w^{13}t-1568w^{12}t^{2}-384w^{11}t^{3}-15456w^{10}t^{4}-4032w^{9}t^{5}-83104w^{8}t^{6}-19840w^{7}t^{7}-425024w^{6}t^{8}-120488w^{5}t^{9}-1615232w^{4}t^{10}-821832w^{3}t^{11}-3403488w^{2}t^{12}-3182088wt^{13}-224t^{14}}{t^{4}(27x^{6}t^{4}-198x^{4}t^{6}+1074x^{2}t^{8}-54xy^{5}t^{4}+558xy^{3}t^{6}-960xyt^{8}-180y^{4}t^{6}-816y^{2}t^{8}+40zw^{9}+8zw^{8}t+272zw^{7}t^{2}+16zw^{6}t^{3}+904zw^{5}t^{4}-88zw^{4}t^{5}+2048zw^{3}t^{6}-504zw^{2}t^{7}+3456zwt^{8}-14w^{10}-4w^{9}t-98w^{8}t^{2}-24w^{7}t^{3}-322w^{6}t^{4}-68w^{5}t^{5}-686w^{4}t^{6}-136w^{3}t^{7}-984w^{2}t^{8}-740wt^{9})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.96.3.z.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 3X^{3}Y^{2}-X^{4}Z-18X^{2}Y^{2}Z+X^{3}Z^{2}+30XY^{2}Z^{2}+2X^{2}Z^{3}-12Y^{2}Z^{3}-2XZ^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.96.3.z.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x+y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x^{3}t-6x^{2}yt+10xy^{2}t-4y^{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
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.
