Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} t - 2 x z t - x w t - y^{2} t + y z t + z^{2} t + z w t $ |
| $=$ | $2 x^{2} t - 3 x y t + y^{2} t - y w t + z w t$ |
| $=$ | $2 x z t + x w t - y z t - 2 y w t + 2 z w t + w^{2} t$ |
| $=$ | $2 x^{2} w - 3 x y w + y^{2} w - y w^{2} + z w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 52 x^{6} + 138 x^{5} z - 8 x^{4} y^{2} + 141 x^{4} z^{2} - 20 x^{3} y^{2} z + 80 x^{3} z^{3} - 8 x^{2} y^{4} + \cdots + 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 |
|---|
| $(0:0:0:0:1)$, $(-1:-2:-2:1:0)$, $(1:1:0:0:0)$, $(-1:0:-2:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2\cdot13^2}\cdot\frac{65183636292120xw^{10}-89580835848564xw^{8}t^{2}+38311146065655xw^{6}t^{4}-2469374199252xw^{4}t^{6}+1144990119780xw^{2}t^{8}-13663220064xt^{10}-6676590726y^{11}+27909352224y^{9}t^{2}-63357437520y^{7}t^{4}+68216177718y^{5}t^{6}-35459224086y^{3}t^{8}+83446462152288yzw^{9}+10897308438642yzw^{7}t^{2}+31373980301853yzw^{5}t^{4}+2118626235882yzw^{3}t^{6}+132521313834yzwt^{8}-82004603611968yw^{10}+70942717435644yw^{8}t^{2}-53635175967870yw^{6}t^{4}+26333227764yw^{4}t^{6}-1256984820702yw^{2}t^{8}+25435435922yt^{10}-55194525581346z^{2}w^{9}+83076700444722z^{2}w^{7}t^{2}-2070482237334z^{2}w^{5}t^{4}+5069554218846z^{2}w^{3}t^{6}-48662061786z^{2}wt^{8}+41207518757232zw^{10}+85059679334382zw^{8}t^{2}+46479933001332zw^{6}t^{4}+5400493248654zw^{4}t^{6}-34650943470zw^{2}t^{8}-15453594234zt^{10}+29122239194496w^{11}+75691295952930w^{9}t^{2}+11485189097091w^{7}t^{4}+3040846746624w^{5}t^{6}-586373755500w^{3}t^{8}-2348799180wt^{10}}{t^{6}(1800540xw^{4}-589215xw^{2}t^{2}+251784xt^{4}+2195874yzw^{3}+386802yzwt^{2}-2449710yw^{4}+407436yw^{2}t^{2}-251784yt^{4}-1260351z^{2}w^{3}+456924z^{2}wt^{2}+1584693zw^{4}+437199zw^{2}t^{2}-75296zt^{4}+935523w^{5}+483684w^{3}t^{2}-163540wt^{4})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
12.72.3.cf.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 52X^{6}-8X^{4}Y^{2}-8X^{2}Y^{4}+138X^{5}Z-20X^{3}Y^{2}Z+2XY^{4}Z+141X^{4}Z^{2}-18X^{2}Y^{2}Z^{2}+Y^{4}Z^{2}+80X^{3}Z^{3}-8XY^{2}Z^{3}+30X^{2}Z^{4}-2Y^{2}Z^{4}+6XZ^{5}+Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
12.72.3.cf.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{5}{6}z^{5}-\frac{5}{6}z^{4}w+\frac{13}{6}z^{3}w^{2}+\frac{1}{3}z^{3}t^{2}+\frac{5}{6}z^{2}w^{3}-\frac{1}{2}z^{2}wt^{2}-\frac{2}{3}zw^{4}-\frac{2}{3}w^{5}+\frac{1}{6}w^{3}t^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{9}{4}z^{19}t-\frac{123}{16}z^{18}wt+\frac{189}{16}z^{17}w^{2}t+\frac{9}{8}z^{17}t^{3}+\frac{819}{16}z^{16}w^{3}t+\frac{15}{16}z^{16}wt^{3}-\frac{507}{16}z^{15}w^{4}t-\frac{159}{16}z^{15}w^{2}t^{3}-\frac{2187}{16}z^{14}w^{5}t-\frac{27}{16}z^{14}w^{3}t^{3}+\frac{999}{16}z^{13}w^{6}t+\frac{549}{16}z^{13}w^{4}t^{3}+\frac{2745}{16}z^{12}w^{7}t-\frac{225}{16}z^{12}w^{5}t^{3}-\frac{1053}{16}z^{11}w^{8}t-\frac{729}{16}z^{11}w^{6}t^{3}-\frac{819}{8}z^{10}w^{9}t+\frac{639}{16}z^{10}w^{7}t^{3}+\frac{21}{2}z^{9}w^{10}t+\frac{117}{16}z^{9}w^{8}t^{3}+54z^{8}w^{11}t-\frac{117}{8}z^{8}w^{9}t^{3}-9z^{7}w^{12}t+\frac{3}{4}z^{7}w^{10}t^{3}-6z^{6}w^{13}t+\frac{3}{2}z^{6}w^{11}t^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z^{5}-\frac{3}{2}z^{3}w^{2}+z^{2}w^{3}$ |
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.
