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 - 6 x^{4} y^{2} + 30 x^{4} z^{2} - 24 x^{3} y^{2} z + 80 x^{3} z^{3} + 9 x^{2} y^{4} + \cdots + 52 z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{7} + 7x^{4} - 8x $ |
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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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2\,\frac{5016xw^{10}-24576xw^{8}t^{2}+32592xw^{6}t^{4}+4428xw^{4}t^{6}-46365xw^{2}t^{8}+19821xt^{10}+24y^{7}t^{4}-312y^{5}t^{6}+1920y^{3}t^{8}+7392yzw^{9}-31584yzw^{7}t^{2}+61632yzw^{5}t^{4}-62514yzw^{3}t^{6}+20190yzwt^{8}+4176yw^{10}-20928yw^{8}t^{2}+60216yw^{6}t^{4}-98271yw^{4}t^{6}+72762yw^{2}t^{8}-25029yt^{10}-7296z^{2}w^{9}+35616z^{2}w^{7}t^{2}-67968z^{2}w^{5}t^{4}+60968z^{2}w^{3}t^{6}-14472z^{2}wt^{8}+2040zw^{10}-7392zw^{8}t^{2}+27120zw^{6}t^{4}-49738zw^{4}t^{6}+38391zw^{2}t^{8}-7305zt^{10}+5784w^{11}-27336w^{9}t^{2}+62040w^{7}t^{4}-76363w^{5}t^{6}+38277w^{3}t^{8}-12300wt^{10}}{t^{6}(21xw^{4}+36xw^{2}t^{2}+21xt^{4}+12yzw^{3}+12yzwt^{2}-9yw^{4}-18yw^{2}t^{2}-21yt^{4}-12z^{2}w^{3}-16z^{2}wt^{2}-10zw^{2}t^{2}-12zt^{4}-w^{3}t^{2}-9wt^{4})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
12.72.3.da.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}-6X^{4}Y^{2}+9X^{2}Y^{4}+6X^{5}Z-24X^{3}Y^{2}Z+18XY^{4}Z+30X^{4}Z^{2}-54X^{2}Y^{2}Z^{2}-72Y^{4}Z^{2}+80X^{3}Z^{3}-60XY^{2}Z^{3}+141X^{2}Z^{4}-24Y^{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.da.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{13}z^{4}-\frac{2}{13}z^{3}w-\frac{11}{52}z^{2}w^{2}+\frac{3}{52}z^{2}t^{2}+\frac{2}{13}zw^{3}+\frac{3}{52}zwt^{2}+\frac{3}{26}w^{4}-\frac{3}{26}w^{2}t^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{165}{228488}z^{9}w^{6}t-\frac{2577}{456976}z^{8}w^{7}t-\frac{6909}{913952}z^{7}w^{8}t+\frac{495}{913952}z^{7}w^{6}t^{3}+\frac{2445}{1827904}z^{6}w^{9}t+\frac{6741}{1827904}z^{6}w^{7}t^{3}+\frac{4503}{281216}z^{5}w^{10}t+\frac{5643}{1827904}z^{5}w^{8}t^{3}+\frac{93777}{7311616}z^{4}w^{11}t-\frac{14211}{1827904}z^{4}w^{9}t^{3}-\frac{50019}{14623232}z^{3}w^{12}t-\frac{7227}{1124864}z^{3}w^{10}t^{3}-\frac{242877}{29246464}z^{2}w^{13}t+\frac{81351}{29246464}z^{2}w^{11}t^{3}-\frac{104691}{29246464}zw^{14}t+\frac{45}{13312}zw^{12}t^{3}-\frac{3669}{7311616}w^{15}t+\frac{10539}{14623232}w^{13}t^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{39}z^{4}+\frac{2}{39}z^{3}w-\frac{5}{52}z^{2}w^{2}-\frac{1}{52}z^{2}t^{2}-\frac{5}{78}zw^{3}-\frac{1}{52}zwt^{2}-\frac{1}{312}w^{4}+\frac{1}{26}w^{2}t^{2}$ |
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.
