Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - y u + z w - 2 w t $ |
| $=$ | $x u + y u + 2 z w + w t$ |
| $=$ | $x z - 2 x t + 3 y z - y t$ |
| $=$ | $5 x z + x t - y z + w u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 5 x^{4} y^{4} + 15 x^{4} y^{2} z^{2} + 9 x^{4} z^{4} + 50 x^{2} y^{4} z^{2} + 135 x^{2} y^{2} z^{4} + \cdots - 25 y^{4} z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -30x^{7} - 225x^{6} + 2070x^{5} - 3375x^{4} + 3750x^{3} - 3435x^{2} + 1650x - 285 $ |
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This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{3}{10}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}u$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{3}{4}w^{2}t^{4}-\frac{3}{10}w^{2}t^{2}u^{2}+\frac{3}{500}w^{2}u^{4}+\frac{2}{5}t^{3}u^{3}-\frac{16}{75}t^{2}u^{4}-\frac{2}{75}tu^{5}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{15}{2}w^{3}t^{16}u^{5}+\frac{15}{2}w^{3}t^{15}u^{6}-\frac{21}{2}w^{3}t^{14}u^{7}-\frac{27}{5}w^{3}t^{13}u^{8}+\frac{109}{25}w^{3}t^{12}u^{9}+\frac{13}{10}w^{3}t^{11}u^{10}-\frac{149}{225}w^{3}t^{10}u^{11}-\frac{818}{5625}w^{3}t^{9}u^{12}+\frac{169}{3750}w^{3}t^{8}u^{13}+\frac{461}{56250}w^{3}t^{7}u^{14}-\frac{77}{56250}w^{3}t^{6}u^{15}-\frac{31}{140625}w^{3}t^{5}u^{16}+\frac{2}{140625}w^{3}t^{4}u^{17}+\frac{1}{468750}w^{3}t^{3}u^{18}+\frac{5}{2}wt^{16}u^{7}-2wt^{15}u^{8}+\frac{27}{5}wt^{14}u^{9}-\frac{26}{5}wt^{13}u^{10}-\frac{23}{30}wt^{12}u^{11}+\frac{596}{375}wt^{11}u^{12}+\frac{182}{3375}wt^{10}u^{13}-\frac{2924}{16875}wt^{9}u^{14}-\frac{239}{33750}wt^{8}u^{15}+\frac{698}{84375}wt^{7}u^{16}+\frac{49}{84375}wt^{6}u^{17}-\frac{62}{421875}wt^{5}u^{18}-\frac{13}{843750}wt^{4}u^{19}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{4}w^{2}t^{4}-\frac{1}{10}w^{2}t^{2}u^{2}+\frac{1}{500}w^{2}u^{4}+\frac{1}{3}t^{4}u^{2}-\frac{1}{5}t^{3}u^{3}-\frac{7}{75}t^{2}u^{4}+\frac{1}{75}tu^{5}$ |
Maps to other modular curves
$j$-invariant map
of degree 60 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{5^3}\cdot\frac{25920xw^{7}+34560xw^{5}u^{2}-39744xw^{3}u^{4}-167040xwu^{6}-51840yw^{7}-60480yw^{5}u^{2}-145152yw^{3}u^{4}-323520ywu^{6}+118125zt^{6}u+826875zt^{4}u^{3}+2505375zt^{2}u^{5}-3375zu^{7}+16875t^{7}u+145125t^{5}u^{3}+473625t^{3}u^{5}+353375tu^{7}}{u(140zt^{6}+80zt^{4}u^{2}-20zt^{2}u^{4}-4zu^{6}+20t^{7}+22t^{5}u^{2}-5t^{3}u^{4}-7tu^{6})}$ |
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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.
