Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ - 2 x t + y w - 2 y t - 2 z w $ |
| $=$ | $8 x w - x t + z w - 2 z t$ |
| $=$ | $16 x^{2} + 15 x y + 4 x z + 4 z^{2}$ |
| $=$ | $15 x^{2} - 15 x y + 15 y^{2} + 4 w^{2} - w t + t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4096 x^{6} - 9216 x^{5} z + 455 x^{4} y^{2} + 6480 x^{4} z^{2} - 170 x^{3} y^{2} z - 1730 x^{3} z^{3} + \cdots + z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -9x^{6} - 60x^{4} - 150x^{2} - 125 $ |
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This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 36 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 3^3\,\frac{495110042247616757144164800xz^{5}-608353657198444514925313200xz^{3}t^{2}-1107228406970719289061936000xzt^{4}+1374401019733142079992040000y^{2}z^{4}+769699882318574204965913400y^{2}z^{2}t^{2}-542478823507494525467414025y^{2}t^{4}-723342358950196104756000000yz^{5}-1236555837855730327904841600yz^{3}t^{2}+161244091052664509292003000yzt^{4}+74130981327485968348008000z^{6}-214893278804036166215590800z^{4}t^{2}-4052251712957291412710400z^{2}t^{4}-57750753703133908064135943w^{6}-8227512731644891631588096w^{5}t+18852858609571582582587328w^{4}t^{2}-29310683269041703970503824w^{3}t^{3}-50164840703930690842846928w^{2}t^{4}-110295775760056644223450160wt^{5}+3083866611748987344085200t^{6}}{7736094410119011830377575xz^{5}+1481393559318327097934400xz^{3}t^{2}-262864240921921691002275xzt^{4}+21475015933330344999875625y^{2}z^{4}-314499161492092650384675y^{2}z^{2}t^{2}-125207085129688063700100y^{2}t^{4}-11302224358596814136812500yz^{5}+1066203258195602306707200yz^{3}t^{2}+785560179792565172937600yzt^{4}+1158296583241968255437625z^{6}+309044150398081621156725z^{4}t^{2}+88070926865917142969325z^{2}t^{4}+21579240003915821101056w^{6}-174154905759797601486848w^{5}t+525069975768168092464384w^{4}t^{2}-684545729073158525505792w^{3}t^{3}+357570032392556941135216w^{2}t^{4}-157609265423280720252260wt^{5}+26863160504135033991300t^{6}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.36.2.bd.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 12z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4096X^{6}+455X^{4}Y^{2}-9216X^{5}Z-170X^{3}Y^{2}Z+6480X^{4}Z^{2}+105X^{2}Y^{2}Z^{2}-1730X^{3}Z^{3}-20XY^{2}Z^{3}+405X^{2}Z^{4}+5Y^{2}Z^{4}-36XZ^{5}+Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.36.2.bd.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{2}w^{3}-\frac{17}{16}w^{2}t+\frac{1}{8}wt^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{273}{160}zw^{8}+\frac{5457}{640}zw^{7}t-\frac{159249}{10240}zw^{6}t^{2}+\frac{9921}{640}zw^{5}t^{3}-\frac{1563}{128}zw^{4}t^{4}+\frac{483}{80}zw^{3}t^{5}-\frac{717}{320}zw^{2}t^{6}+\frac{57}{160}zwt^{7}-\frac{3}{160}zt^{8}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{10}w^{3}+\frac{33}{80}w^{2}t-\frac{9}{20}wt^{2}+\frac{1}{20}t^{3}$ |
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.
