Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ - x t^{2} + y^{2} t $ |
| $=$ | $y w^{2} + y w t + z w t$ |
| $=$ | $y w t + y t^{2} + z t^{2}$ |
| $=$ | $x w t + x t^{2} + y z t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y + 5 x^{4} z - x^{2} y^{2} z - 6 x^{2} y z^{2} - 5 x^{2} z^{3} + y^{2} z^{3} + y z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 2x^{6} - x^{4} + 2x^{2} $ |
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This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:1:0:0)$, $(1:-1:1:0:1)$, $(0:0:0:-1:1)$, $(1:0:0:0:0)$, $(0:0:-1:1:0)$, $(0:0:1:1:0)$, $(1:1:-1:0:1)$, $(0:0:0:0:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{x^{11}-705x^{10}t+170015x^{9}t^{2}-15035355x^{8}t^{3}+254385970x^{7}t^{4}-2560733026x^{6}t^{5}-4296636730x^{5}t^{6}-85385769230x^{4}t^{7}-196268373347x^{3}t^{8}-271804879117x^{2}t^{9}+320871972160xwt^{9}+320871972235xt^{10}+25600z^{10}t-51200z^{8}w^{2}t-307200z^{8}t^{3}+1408024z^{6}w^{2}t^{3}-5093914z^{6}wt^{4}+34899252z^{6}t^{5}-181832860z^{4}w^{2}t^{5}+374770690z^{4}wt^{6}-751038298z^{4}t^{7}+11644965696z^{2}w^{2}t^{7}+73670355200z^{2}wt^{8}+239921319936z^{2}t^{9}-25w^{11}+25675w^{10}t-154475w^{9}t^{2}-199499w^{8}t^{3}+6543608w^{7}t^{4}+99804538w^{6}t^{5}-556106258w^{5}t^{6}-8994220992w^{4}t^{7}-60160972083w^{3}t^{8}-163974757569w^{2}t^{9}-112158658639wt^{10}}{t(x^{10}+37x^{9}t+590x^{8}t^{2}+5410x^{7}t^{3}+32513x^{6}t^{4}+140649x^{5}t^{5}+474800x^{4}t^{6}+1439696x^{3}t^{7}+7013040x^{2}t^{8}-7372964xwt^{8}-7372964xt^{9}-z^{6}wt^{3}-31z^{6}t^{4}-417z^{4}w^{2}t^{4}-4034z^{4}wt^{5}-24435z^{4}t^{6}-93104z^{2}w^{2}t^{6}-566430z^{2}wt^{7}-1709306z^{2}t^{8}+w^{7}t^{3}+467w^{6}t^{4}+1894w^{5}t^{5}+114800w^{4}t^{6}+288975w^{3}t^{7}+1992057w^{2}t^{8}+1816454wt^{9})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
$X_0(40)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{4}Y+5X^{4}Z-X^{2}Y^{2}Z-6X^{2}YZ^{2}-5X^{2}Z^{3}+Y^{2}Z^{3}+YZ^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
$X_0(40)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -y^{2}wt-3y^{2}t^{2}+wt^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
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.
