Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 2 x w t + y w t + y t^{2} - z t^{2} $ |
| $=$ | $2 x w^{2} + y w^{2} + y w t - z w t$ |
| $=$ | $2 x y w + y^{2} w + y^{2} t - y z t$ |
| $=$ | $2 x^{2} w + x y w + x y t - x z t$ |
| $=$ | $\cdots$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{7} + 96 x^{6} z - 33 x^{5} y^{2} + 145 x^{5} z^{2} + 15 x^{4} y^{2} z + 120 x^{4} z^{3} + \cdots + 3 y^{2} z^{5} $ |
![Copy content]()
![Toggle raw display]()
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -3x^{7} - 27x^{5} - 75x^{3} + 33x^{2} - 60x + 132 $ |
![Copy content]()
![Toggle raw display]()
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle w^{2}t+\frac{4}{3}wt^{2}+t^{3}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -11zw^{11}-39zw^{10}t-\frac{215}{3}zw^{9}t^{2}-\frac{1820}{27}zw^{8}t^{3}-\frac{535}{27}zw^{7}t^{4}+\frac{121}{3}zw^{6}t^{5}+\frac{1751}{27}zw^{5}t^{6}+55zw^{4}t^{7}+\frac{820}{27}zw^{3}t^{8}+\frac{40}{3}zw^{2}t^{9}+4zwt^{10}+zt^{11}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w^{3}+\frac{4}{3}w^{2}t+wt^{2}$ |
Maps to other modular curves
$j$-invariant map
of degree 60 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^8\cdot11^8}\cdot\frac{9671731426781532900xyz^{7}-39390912258395454900xyz^{5}t^{2}-13835765085908040000xyz^{3}t^{4}-699644415481096084200xyzt^{6}-4034444858415406215xz^{8}+4155155689681800540xz^{6}t^{2}-14581715327492936040xz^{4}t^{4}-168012672506712529560xz^{2}t^{6}+2511252566836791360xt^{8}+1409748838709322375y^{2}z^{7}-3109070819148020100y^{2}z^{5}t^{2}+17311700853474409800y^{2}z^{3}t^{4}+166533296082770885400y^{2}zt^{6}+1139190980775210yz^{8}+21663229144213989090yz^{6}t^{2}+52724160417034283760yz^{4}t^{4}+997098627390838368840yz^{2}t^{6}+8740096703651096960yt^{8}+1409179243218934770z^{9}-10619982910634835870z^{7}t^{2}-16613761997938655880z^{5}t^{4}-350292489777043618320z^{3}t^{6}+25887435892446915072zw^{8}+254247540798316944384zw^{7}t+1062468647318610297216zw^{6}t^{2}+2806066923379108823808zw^{5}t^{3}+5072218689275101916640zw^{4}t^{4}+7006542493471414757568zw^{3}t^{5}+1171332079330271266056zw^{2}t^{6}+1589718161553078215024zwt^{7}-9577060879061270848zt^{8}}{7170xt^{8}-6505yt^{8}-24057zw^{8}+43011zw^{7}t-33291zw^{6}t^{2}+12312zw^{5}t^{3}+16875zw^{4}t^{4}-32625zw^{3}t^{5}+26625zw^{2}t^{6}-2875zwt^{7}+4115zt^{8}}$ |
Hi
|
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.
