Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x^{2} + y z $ |
| $=$ | $2 x w - x t + z u$ |
| $=$ | $ - x u + 2 y w - y t$ |
| $=$ | $3 x^{2} - 3 y z - 6 z^{2} + 4 w^{2} - 2 w t + t^{2} + u^{2}$ |
| $=$ | $\cdots$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} - 4 x^{4} y^{2} + 2 x^{4} z^{2} + 4 x^{2} y^{4} - 8 x^{2} y^{2} z^{2} + x^{2} z^{4} - 6 y^{2} z^{4} $ |
![Copy content]()
![Toggle raw display]()
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 2x^{8} + 240x^{4} + 2592 $ |
![Copy content]()
![Toggle raw display]()
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^2\cdot3^3\,\frac{508xt^{7}u-9629xt^{5}u^{3}-28604xt^{3}u^{5}-14540xtu^{7}+6967yt^{8}+15877yt^{6}u^{2}-3078yt^{4}u^{4}-21992yt^{2}u^{6}-9992yu^{8}+192zw^{8}-192zw^{4}u^{4}+1824zw^{2}u^{6}+3072zt^{8}+6459zt^{6}u^{2}+3206zt^{4}u^{4}+6912zt^{2}u^{6}+5688zu^{8}}{17xt^{7}u+497xt^{5}u^{3}+2324xt^{3}u^{5}+2348xtu^{7}+2yt^{8}+188yt^{6}u^{2}+1518yt^{4}u^{4}+2984yt^{2}u^{6}+1652yu^{8}-324zw^{2}u^{6}-15zt^{6}u^{2}-335zt^{4}u^{4}-1248zt^{2}u^{6}-972zu^{8}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.72.3.og.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{12}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{6}u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{6}-4X^{4}Y^{2}+4X^{2}Y^{4}+2X^{4}Z^{2}-8X^{2}Y^{2}Z^{2}+X^{2}Z^{4}-6Y^{2}Z^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.72.3.og.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{4}y^{4}t+\frac{1}{2}y^{4}u+\frac{1}{4}y^{3}u^{2}+\frac{1}{288}y^{2}t^{3}-\frac{1}{48}y^{2}tu^{2}+\frac{1}{72}y^{2}u^{3}+\frac{1}{144}yu^{4}-\frac{1}{1728}tu^{4}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -105y^{17}u^{3}+21y^{16}u^{4}+\frac{35}{24}y^{15}t^{2}u^{3}-\frac{37}{8}y^{15}tu^{4}-\frac{35}{2}y^{15}u^{5}-\frac{7}{24}y^{14}t^{2}u^{4}+\frac{5}{8}y^{14}tu^{5}+\frac{7}{2}y^{14}u^{6}+\frac{37}{1728}y^{13}t^{3}u^{4}+\frac{35}{216}y^{13}t^{2}u^{5}-\frac{2729}{3456}y^{13}tu^{6}-\frac{1085}{864}y^{13}u^{7}-\frac{5}{1728}y^{12}t^{3}u^{5}-\frac{7}{216}y^{12}t^{2}u^{6}+\frac{35}{432}y^{12}tu^{7}+\frac{217}{864}y^{12}u^{8}+\frac{221}{82944}y^{11}t^{3}u^{6}+\frac{35}{5184}y^{11}t^{2}u^{7}-\frac{1177}{20736}y^{11}tu^{8}-\frac{385}{7776}y^{11}u^{9}-\frac{5}{20736}y^{10}t^{3}u^{7}-\frac{7}{5184}y^{10}t^{2}u^{8}+\frac{5}{1152}y^{10}tu^{9}+\frac{77}{7776}y^{10}u^{10}+\frac{367}{2985984}y^{9}t^{3}u^{8}+\frac{35}{279936}y^{9}t^{2}u^{9}-\frac{367}{165888}y^{9}tu^{10}-\frac{35}{31104}y^{9}u^{11}-\frac{5}{746496}y^{8}t^{3}u^{9}-\frac{7}{279936}y^{8}t^{2}u^{10}+\frac{5}{41472}y^{8}tu^{11}+\frac{7}{31104}y^{8}u^{12}+\frac{805}{322486272}y^{7}t^{3}u^{10}+\frac{35}{40310784}y^{7}t^{2}u^{11}-\frac{7985}{161243136}y^{7}tu^{12}-\frac{35}{2519424}y^{7}u^{13}-\frac{5}{80621568}y^{6}t^{3}u^{11}-\frac{7}{40310784}y^{6}t^{2}u^{12}+\frac{35}{20155392}y^{6}tu^{13}+\frac{7}{2519424}y^{6}u^{14}+\frac{73}{3869835264}y^{5}t^{3}u^{12}-\frac{65}{107495424}y^{5}tu^{14}-\frac{35}{483729408}y^{5}u^{15}+\frac{5}{483729408}y^{4}tu^{15}+\frac{7}{483729408}y^{4}u^{16}-\frac{73}{23219011584}y^{3}tu^{16}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{4}y^{4}t+\frac{1}{2}y^{4}u-\frac{1}{24}y^{3}u^{2}-\frac{1}{288}y^{2}t^{3}+\frac{1}{48}y^{2}tu^{2}+\frac{1}{72}y^{2}u^{3}-\frac{1}{864}yu^{4}+\frac{1}{1728}tu^{4}$ |
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.
