Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - x u + z t $ |
| $=$ | $3 y z + w u$ |
| $=$ | $3 x y + w t$ |
| $=$ | $ - 3 x t - x u + y w - z t + z u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 256 x^{4} y^{4} - 6240 x^{4} y^{2} z^{2} + 38025 x^{4} z^{4} - 252 x^{2} y^{6} + 5490 x^{2} y^{4} z^{2} + \cdots + 10125 y^{4} z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 15x^{8} - 360x^{6} + 2970x^{4} - 16200x^{2} + 30375 $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{4}{15}u$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{4}{663}zwt^{2}+\frac{1}{663}zwtu+\frac{1}{1020}t^{3}u+\frac{7}{13260}t^{2}u^{2}-\frac{1}{2652}tu^{3}+\frac{1}{13260}u^{4}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{161}{2255067000}zt^{12}u^{3}+\frac{88943}{351790452000}zt^{11}u^{4}-\frac{2673221}{6859913814000}zt^{10}u^{5}+\frac{63343177}{178357759164000}zt^{9}u^{6}-\frac{31251877}{144915679320750}zt^{8}u^{7}+\frac{8235101}{89178879582000}zt^{7}u^{8}-\frac{5525657}{193220905761000}zt^{6}u^{9}+\frac{2484347}{386441811522000}zt^{5}u^{10}-\frac{597751}{579662717283000}zt^{4}u^{11}+\frac{261373}{2318650869132000}zt^{3}u^{12}-\frac{517}{68195613798000}zt^{2}u^{13}+\frac{553}{2318650869132000}ztu^{14}-\frac{2}{56376675}wt^{11}u^{4}+\frac{5893}{65960709750}wt^{10}u^{5}-\frac{89203}{857489226750}wt^{9}u^{6}+\frac{411314}{5573679973875}wt^{8}u^{7}-\frac{2559314}{72457839660375}wt^{7}u^{8}+\frac{287941}{24152613220125}wt^{6}u^{9}-\frac{69439}{24152613220125}wt^{5}u^{10}+\frac{35474}{72457839660375}wt^{4}u^{11}-\frac{4088}{72457839660375}wt^{3}u^{12}+\frac{577}{144915679320750}wt^{2}u^{13}-\frac{19}{144915679320750}wtu^{14}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{663}zwt^{2}+\frac{1}{2652}zwtu-\frac{1}{255}t^{3}u+\frac{31}{9945}t^{2}u^{2}-\frac{1}{1105}tu^{3}+\frac{1}{9945}u^{4}$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{5^2}\cdot\frac{64646400000z^{2}w^{8}-382809600000z^{2}w^{6}u^{2}-147020136000z^{2}w^{4}u^{4}+29001382875z^{2}w^{2}u^{6}+19973369775z^{2}u^{8}-69068800000w^{10}+393182720000w^{8}u^{2}+246283384000w^{6}u^{4}+233982775w^{4}u^{6}-26792930235w^{2}u^{8}+2569753885184t^{10}-7365207209280t^{9}u-19198146501584t^{8}u^{2}+7012681067720t^{7}u^{3}+15808779838641t^{6}u^{4}-8240072606760t^{5}u^{5}-2479270872261t^{4}u^{6}+3495549907727t^{3}u^{7}-1509587183693t^{2}u^{8}+323891937367tu^{9}-32297715853u^{10}}{24000z^{2}w^{4}u^{4}-9000z^{2}w^{2}u^{6}+615z^{2}u^{8}-8000w^{6}u^{4}+6200w^{4}u^{6}-1085w^{2}u^{8}-122487552t^{10}-77267520t^{9}u+75461760t^{8}u^{2}+17233920t^{7}u^{3}-13076664t^{6}u^{4}+33712t^{5}u^{5}+149300t^{4}u^{6}+45647t^{3}u^{7}-740t^{2}u^{8}+983tu^{9}+474u^{10}}$ |
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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.
