Embedded model Embedded model in $\mathbb{P}^{5}$
$ 0 $ | $=$ | $ x^{2} + x w - z t $ |
| $=$ | $x^{2} - x w - y t$ |
| $=$ | $x y - x z + y w + z w$ |
| $=$ | $x z + 2 x t + y w - 2 z w$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 36 x^{8} + 864 x^{6} y^{2} - 72 x^{6} y z + 888 x^{6} z^{2} + 5832 x^{4} y^{4} - 2376 x^{4} y^{3} z + \cdots + 132496 z^{8} $ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ -6x^{8} + 48x^{7} - 384x^{6} - 672x^{5} + 1200x^{4} + 1344x^{3} - 1536x^{2} - 384x - 96 $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle u$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Birational map from embedded model to Weierstrass model:
$\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{3}zt-2w^{2}-\frac{2}{3}wt+\frac{2}{3}t^{2}$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{8}{3}zw^{5}t^{2}+\frac{16}{9}zw^{4}t^{3}-\frac{8}{3}zw^{4}t^{2}u-\frac{16}{9}zw^{3}t^{3}u-\frac{40}{3}w^{7}t-\frac{32}{3}w^{6}t^{2}+\frac{40}{3}w^{6}tu+\frac{40}{9}w^{5}t^{3}+\frac{32}{3}w^{5}t^{2}u+\frac{32}{9}w^{4}t^{4}-\frac{40}{9}w^{4}t^{3}u-\frac{32}{9}w^{3}t^{4}u$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}zt-w^{2}+\frac{1}{3}wt+\frac{2}{3}t^{2}$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3^6}\cdot\frac{401444028ywt^{6}u+206241552ywt^{4}u^{3}+8188128ywt^{2}u^{5}-839808ywu^{7}-223513955yt^{8}-388311750yt^{6}u^{2}-40640292yt^{4}u^{4}+5493744yt^{2}u^{6}+104976yu^{8}+401444028zwt^{6}u+206241552zwt^{4}u^{3}+8188128zwt^{2}u^{5}-839808zwu^{7}-223513955zt^{8}-388311750zt^{6}u^{2}-40640292zt^{4}u^{4}+5493744zt^{2}u^{6}+104976zu^{8}-65533404wt^{7}u-791788608wt^{5}u^{3}-204563232wt^{3}u^{5}-9517824wtu^{7}-435727868t^{9}+919317966t^{7}u^{2}+499118760t^{5}u^{4}+52056432t^{3}u^{6}+699840tu^{8}}{t^{6}(36ywu-35yt^{2}-18yu^{2}+36zwu-35zt^{2}-18zu^{2}+132wtu-572t^{3}-66tu^{2})}$ |
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.