Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ y w + z^{2} + z u $ |
| $=$ | $x^{2} - x y + x w + z^{2} + w^{2}$ |
| $=$ | $x^{2} + x w + y w - 2 z^{2} + z u + w^{2}$ |
| $=$ | $3 x^{2} + 3 x y + 3 y^{2} - t^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{8} - 2787 x^{6} y^{2} - 684 x^{6} z^{2} + 211600 x^{4} y^{4} - 135618 x^{4} y^{2} z^{2} + \cdots + 12321 z^{8} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 7x^{4} + 20 $ |
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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. The following are the known rational points on this modular curve (one row per $j$-invariant).
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3^3}\cdot\frac{4991256617582519389xt^{8}-83436465360659659584xt^{6}u^{2}-1165214488871915974800xt^{4}u^{4}-975573712915659575424xt^{2}u^{6}-8455545316865528064xu^{8}-7399788706115367436yt^{8}-68183158027623103332yt^{6}u^{2}-207428305574687798160yt^{4}u^{4}-42352542650741361984yt^{2}u^{6}+5923371433886390016yu^{8}-291466204195106117688zwt^{6}u-3758810886287480245824zwt^{4}u^{3}-3290568832208558146176zwt^{2}u^{5}-69259781291826161664zwu^{7}-29558127100005868056wt^{6}u^{2}-238104013331319041088wt^{4}u^{4}-132533372503834372224wt^{2}u^{6}+9259572006041131008wu^{8}}{478459054579968xt^{6}u^{2}-2745723117473280xt^{4}u^{4}+8234066507779763xt^{2}u^{6}-181231681174244xu^{8}-30126520202688yt^{8}+148562236577664yt^{6}u^{2}-248446969968960yt^{4}u^{4}+3052846660548yt^{2}u^{6}+126958406933436yu^{8}+1603657844635392zwt^{6}u-8752866333200640zwt^{4}u^{3}+28661033224180224zwt^{2}u^{5}-1484477479677344zwu^{7}+140471585323776wt^{6}u^{2}-891961296033024wt^{4}u^{4}+665465256775368wt^{2}u^{6}+198464763503968wu^{8}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
12.72.3.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 9X^{8}-2787X^{6}Y^{2}+211600X^{4}Y^{4}+634557X^{2}Y^{6}+1979649Y^{8}-684X^{6}Z^{2}-135618X^{4}Y^{2}Z^{2}+92694X^{2}Y^{4}Z^{2}+988056Y^{6}Z^{2}+13662X^{4}Z^{4}+54957X^{2}Y^{2}Z^{4}+351486Y^{4}Z^{4}-25308X^{2}Z^{6}+67896Y^{2}Z^{6}+12321Z^{8} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
12.72.3.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{3}{130}z^{3}-\frac{1}{10}z^{2}w+\frac{3}{65}z^{2}u+\frac{1}{65}zwu+\frac{3}{130}zu^{2}-\frac{1}{130}wu^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{2387}{71402500}z^{12}-\frac{59}{1373125}z^{11}w+\frac{54}{1373125}z^{11}t-\frac{2289}{17850625}z^{11}u+\frac{237}{2746250}z^{10}wt-\frac{304}{3570125}z^{10}wu+\frac{2466}{17850625}z^{10}tu-\frac{3331}{17850625}z^{10}u^{2}+\frac{1551}{7140250}z^{9}wtu-\frac{16}{3570125}z^{9}wu^{2}+\frac{2934}{17850625}z^{9}tu^{2}-\frac{2367}{17850625}z^{9}u^{3}+\frac{1089}{7140250}z^{8}wtu^{2}+\frac{264}{3570125}z^{8}wu^{3}+\frac{1098}{17850625}z^{8}tu^{3}-\frac{2121}{35701250}z^{8}u^{4}+\frac{99}{7140250}z^{7}wtu^{3}+\frac{738}{17850625}z^{7}wu^{4}-\frac{198}{17850625}z^{7}tu^{4}-\frac{99}{3570125}z^{7}u^{5}+\frac{171}{35701250}z^{6}wtu^{4}+\frac{192}{17850625}z^{6}wu^{5}-\frac{18}{3570125}z^{6}tu^{5}-\frac{187}{17850625}z^{6}u^{6}+\frac{369}{35701250}z^{5}wtu^{5}+\frac{8}{1373125}z^{5}wu^{6}+\frac{18}{17850625}z^{5}tu^{6}-\frac{33}{17850625}z^{5}u^{7}-\frac{57}{35701250}z^{4}wtu^{6}+\frac{8}{17850625}z^{4}wu^{7}-\frac{18}{17850625}z^{4}tu^{7}-\frac{7}{14280500}z^{4}u^{8}+\frac{21}{35701250}z^{3}wtu^{7}+\frac{1}{3570125}z^{3}wu^{8}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{7}{130}z^{3}-\frac{1}{10}z^{2}w-\frac{3}{65}z^{2}u+\frac{1}{65}zwu+\frac{1}{130}zu^{2}-\frac{1}{130}wu^{2}$ |
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.
