Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x^{2} + z w $ |
| $=$ | $x^{2} + y w + y u$ |
| $=$ | $x^{2} - 2 w t + t v$ |
| $=$ | $x w + x u - y t + z t + t^{2}$ |
| $=$ | $\cdots$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} y - x^{6} z + x^{3} y^{3} z + 2 x^{3} y^{2} z^{2} + x^{3} y z^{3} - y^{4} z^{3} + y^{3} z^{4} $ |
![Copy content]()
![Toggle raw display]()
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:1/2:0:2:1)$, $(0:0:-1:0:1:0:0)$, $(0:0:1:0:0:0:0)$, $(0:1:1:0:0:0:0)$, $(0:0:0:1/2:0:-1/2:1)$, $(0:0:0:-1/3:0:1/3:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 108 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^2}\cdot\frac{2112580842927345xtuv^{6}+677966670415790xtv^{7}+34359738368z^{9}+1030792151040z^{3}v^{6}-265516611010560wu^{8}+2226302767595520wu^{7}v-2582161827502080wu^{6}v^{2}-3364834630329600wu^{5}v^{3}+4844042534447520wu^{4}v^{4}+2627324009157420wu^{3}v^{5}-6110308072191035wu^{2}v^{6}-5828847421102935wuv^{7}-664050976376750wv^{8}-1030792151040t^{6}u^{2}v-182713133301760t^{6}uv^{2}-798583166074880t^{6}v^{3}-4291246777989120t^{3}u^{3}v^{3}+2022277851621120t^{3}u^{2}v^{4}+2507750224196640t^{3}uv^{5}+1368302846644060t^{3}v^{6}+184774717603840u^{9}-388843556044800u^{8}v+643812344494080u^{7}v^{2}-2482737671991040u^{6}v^{3}+888448206724320u^{5}v^{4}+1266337342549380u^{4}v^{5}+6240612261375605u^{3}v^{6}-3698089524814625u^{2}v^{7}-2654314119896760uv^{8}}{2668184809xtuv^{6}+640209726xtv^{7}-268959744wu^{8}+3468689408wu^{7}v-3793967104wu^{6}v^{2}-4789812480wu^{5}v^{3}+5823777952wu^{4}v^{4}+4100891596wu^{3}v^{5}-8337179875wu^{2}v^{6}-6111716639wuv^{7}-640209726wv^{8}-187170816t^{6}uv^{2}-969932800t^{6}v^{3}-4156143616t^{3}u^{3}v^{3}+933031680t^{3}u^{2}v^{4}+3764699936t^{3}uv^{5}+1280419452t^{3}v^{6}+187170816u^{9}-258473984u^{8}v-1831225344u^{7}v^{2}+4831807744u^{6}v^{3}-8236038688u^{5}v^{4}+6365740900u^{4}v^{5}+6725431373u^{3}v^{6}-5223573913u^{2}v^{7}-2560838904uv^{8}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
30.108.7.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{6}Y-X^{6}Z+X^{3}Y^{3}Z+2X^{3}Y^{2}Z^{2}+X^{3}YZ^{3}-Y^{4}Z^{3}+Y^{3}Z^{4} $ |
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.
