Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} t + x z t + x w t - y^{2} t - y w t + z w t $ |
| $=$ | $x^{2} y + x y z + x y w - y^{3} - y^{2} w + y z w$ |
| $=$ | $x z t + x w t - 2 y z t + y w t - z^{2} t + w^{2} t$ |
| $=$ | $x^{3} + x^{2} z - x y^{2} - x y w - x w^{2} + y^{2} w + y w^{2} - z w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} - 6 x^{5} z + 2 x^{4} y^{2} + 30 x^{4} z^{2} - 8 x^{3} y^{2} z - 80 x^{3} z^{3} + x^{2} y^{4} + \cdots + 52 z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 3x^{7} - 21x^{4} - 24x $ |
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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 |
|---|
| $(0:0:0:0:1)$, $(-1/2:-1:1/2:1:0)$, $(1:1:0:0:0)$, $(-1/2:0:1/2:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2\,\frac{406296xw^{10}+663552xw^{8}t^{2}+293328xw^{6}t^{4}-13284xw^{4}t^{6}-46365xw^{2}t^{8}-6607xt^{10}+216y^{7}t^{4}+936y^{5}t^{6}+1920y^{3}t^{8}-598752yzw^{9}-852768yzw^{7}t^{2}-554688yzw^{5}t^{4}-187542yzw^{3}t^{6}-20190yzwt^{8}+338256yw^{10}+565056yw^{8}t^{2}+541944yw^{6}t^{4}+294813yw^{4}t^{6}+72762yw^{2}t^{8}+8343yt^{10}-590976z^{2}w^{9}-961632z^{2}w^{7}t^{2}-611712z^{2}w^{5}t^{4}-182904z^{2}w^{3}t^{6}-14472z^{2}wt^{8}-165240zw^{10}-199584zw^{8}t^{2}-244080zw^{6}t^{4}-149214zw^{4}t^{6}-38391zw^{2}t^{8}-2435zt^{10}+468504w^{11}+738072w^{9}t^{2}+558360w^{7}t^{4}+229089w^{5}t^{6}+38277w^{3}t^{8}+4100wt^{10}}{t^{6}(63xw^{4}-36xw^{2}t^{2}+7xt^{4}-36yzw^{3}+12yzwt^{2}-27yw^{4}+18yw^{2}t^{2}-7yt^{4}-36z^{2}w^{3}+16z^{2}wt^{2}-10zw^{2}t^{2}+4zt^{4}+w^{3}t^{2}-3wt^{4})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
12.72.3.ce.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{6}+2X^{4}Y^{2}+X^{2}Y^{4}-6X^{5}Z-8X^{3}Y^{2}Z-2XY^{4}Z+30X^{4}Z^{2}+18X^{2}Y^{2}Z^{2}-8Y^{4}Z^{2}-80X^{3}Z^{3}-20XY^{2}Z^{3}+141X^{2}Z^{4}+8Y^{2}Z^{4}-138XZ^{5}+52Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
12.72.3.ce.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{33}z^{4}+\frac{2}{33}z^{3}w+\frac{7}{44}z^{2}w^{2}-\frac{1}{132}z^{2}t^{2}-\frac{4}{33}zw^{3}+\frac{1}{132}zwt^{2}+\frac{1}{66}w^{4}+\frac{1}{66}w^{2}t^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{27}{117128}z^{9}w^{6}t-\frac{1101}{234256}z^{8}w^{7}t+\frac{717}{117128}z^{7}w^{8}t+\frac{27}{468512}z^{7}w^{6}t^{3}+\frac{579}{468512}z^{6}w^{9}t-\frac{1047}{937024}z^{6}w^{7}t^{3}-\frac{29037}{1874048}z^{5}w^{10}t+\frac{2619}{1874048}z^{5}w^{8}t^{3}+\frac{51207}{3748096}z^{4}w^{11}t+\frac{9387}{3748096}z^{4}w^{9}t^{3}+\frac{10695}{3748096}z^{3}w^{12}t-\frac{9033}{3748096}z^{3}w^{10}t^{3}-\frac{8049}{937024}z^{2}w^{13}t-\frac{1647}{1874048}z^{2}w^{11}t^{3}+\frac{3603}{937024}zw^{14}t+\frac{1125}{937024}zw^{12}t^{3}-\frac{129}{234256}w^{15}t-\frac{123}{468512}w^{13}t^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{2}{33}z^{4}-\frac{4}{33}z^{3}w+\frac{2}{11}z^{2}w^{2}+\frac{1}{66}z^{2}t^{2}+\frac{1}{12}zw^{3}-\frac{1}{66}zwt^{2}-\frac{5}{66}w^{4}-\frac{1}{33}w^{2}t^{2}$ |
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.
