Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} t - x z t - 2 x w t - y^{2} t + y w t + z w t + w^{2} t $ |
| $=$ | $2 x^{2} t - 3 x y t + y^{2} t - y z t + z w t$ |
| $=$ | $x z t + 2 x w t - 2 y z t - y w t + z^{2} t + 2 z w t$ |
| $=$ | $x^{2} y - x y z - 2 x y w - y^{3} + y^{2} w + y z w + y w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 6 x^{5} z + 4 x^{4} y^{2} + 30 x^{4} z^{2} + 16 x^{3} y^{2} z + 80 x^{3} z^{3} + 4 x^{2} y^{4} + \cdots + 52 z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -6x^{7} - 42x^{4} + 48x $ |
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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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2\,\frac{13001472xw^{10}+10616832xw^{8}t^{2}+2346624xw^{6}t^{4}-53136xw^{4}t^{6}-92730xw^{2}t^{8}-6607xt^{10}+1728y^{7}t^{4}+3744y^{5}t^{6}+3840y^{3}t^{8}-19160064yzw^{9}-13644288yzw^{7}t^{2}-4437504yzw^{5}t^{4}-750168yzw^{3}t^{6}-40380yzwt^{8}-8335872yw^{10}-4603392yw^{8}t^{2}-101952yw^{6}t^{4}+429084yw^{4}t^{6}+105144yw^{2}t^{8}+8343yt^{10}+18911232z^{2}w^{9}+15386112z^{2}w^{7}t^{2}+4893696z^{2}w^{5}t^{4}+731616z^{2}w^{3}t^{6}+28944z^{2}wt^{8}+43110144zw^{10}+33965568zw^{8}t^{2}+11740032zw^{6}t^{4}+2060088zw^{4}t^{6}+134670zw^{2}t^{8}+2435zt^{10}+9206784w^{11}+6770304w^{9}t^{2}+2379456w^{7}t^{4}+412116w^{5}t^{6}+29172w^{3}t^{8}-1665wt^{10}}{t^{6}(252xw^{4}-72xw^{2}t^{2}+7xt^{4}-144yzw^{3}+24yzwt^{2}-252yw^{4}+60yw^{2}t^{2}-7yt^{4}+144z^{2}w^{3}-32z^{2}wt^{2}+288zw^{4}-44zw^{2}t^{2}-4zt^{4}+144w^{5}-14w^{3}t^{2}-wt^{4})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.72.3.qp.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{6}+4X^{4}Y^{2}+4X^{2}Y^{4}+6X^{5}Z+16X^{3}Y^{2}Z+8XY^{4}Z+30X^{4}Z^{2}+36X^{2}Y^{2}Z^{2}-32Y^{4}Z^{2}+80X^{3}Z^{3}+40XY^{2}Z^{3}+141X^{2}Z^{4}+16Y^{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
24.72.3.qp.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{2}{3}z^{5}-\frac{2}{3}z^{4}w+\frac{5}{6}z^{3}w^{2}-\frac{1}{12}z^{3}t^{2}+\frac{13}{6}z^{2}w^{3}-\frac{5}{6}zw^{4}+\frac{1}{4}zw^{2}t^{2}-\frac{5}{6}w^{5}-\frac{1}{6}w^{3}t^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 6z^{13}w^{6}t+9z^{12}w^{7}t-54z^{11}w^{8}t+\frac{3}{4}z^{11}w^{6}t^{3}-\frac{21}{2}z^{10}w^{9}t+\frac{3}{8}z^{10}w^{7}t^{3}+\frac{819}{8}z^{9}w^{10}t-\frac{117}{16}z^{9}w^{8}t^{3}+\frac{1053}{16}z^{8}w^{11}t+\frac{117}{32}z^{8}w^{9}t^{3}-\frac{2745}{16}z^{7}w^{12}t+\frac{639}{32}z^{7}w^{10}t^{3}-\frac{999}{16}z^{6}w^{13}t-\frac{729}{32}z^{6}w^{11}t^{3}+\frac{2187}{16}z^{5}w^{14}t-\frac{225}{32}z^{5}w^{12}t^{3}+\frac{507}{16}z^{4}w^{15}t+\frac{549}{32}z^{4}w^{13}t^{3}-\frac{819}{16}z^{3}w^{16}t-\frac{27}{32}z^{3}w^{14}t^{3}-\frac{189}{16}z^{2}w^{17}t-\frac{159}{32}z^{2}w^{15}t^{3}+\frac{123}{16}zw^{18}t+\frac{15}{32}zw^{16}t^{3}+\frac{9}{4}w^{19}t+\frac{9}{16}w^{17}t^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z^{3}w^{2}-\frac{3}{2}z^{2}w^{3}+\frac{1}{2}w^{5}$ |
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.
