Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x w + x t + y w + 2 y t + z w - 2 z t $ |
| $=$ | $3 x w - x t - y w + 2 y t$ |
| $=$ | $5 x^{2} + 5 x z + 5 z^{2} + w^{2} + w t - t^{2}$ |
| $=$ | $4 x^{2} + 4 x y - 5 x z - 4 y^{2}$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{6} + 15 x^{5} z + 13 x^{4} y^{2} + 36 x^{3} y^{2} z - 25 x^{3} z^{3} + 12 x^{2} y^{2} z^{2} + \cdots - 5 z^{6} $ |
![Copy content]()
![Toggle raw display]()
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{6} + 20x^{4} - 150x^{2} + 375 $ |
![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 36 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^6\,\frac{569462400xz^{3}t^{2}+111963600xzt^{4}+545734800y^{2}z^{2}t^{2}+282441600y^{2}t^{4}+23727600yz^{3}t^{2}-285465600yzt^{4}-250622775z^{6}-5931900z^{4}t^{2}+199130400z^{2}t^{4}-3904069w^{6}-2263248w^{5}t+5975424w^{4}t^{2}+20561328w^{3}t^{3}-5975424w^{2}t^{4}+25170480wt^{5}-19095120t^{6}}{5272800xz^{3}t^{2}+10356400xzt^{4}+5053100y^{2}z^{2}t^{2}+3353600y^{2}t^{4}+219700yz^{3}t^{2}+9233600yzt^{4}-9282325z^{6}-54925z^{4}t^{2}-2883000z^{2}t^{4}-140608w^{6}-64896w^{5}t+352768w^{4}t^{2}-1046784w^{3}t^{3}-352768w^{2}t^{4}+1998720wt^{5}-891200t^{6}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.36.2.z.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{5}{4}z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 5X^{6}+13X^{4}Y^{2}+15X^{5}Z+36X^{3}Y^{2}Z+12X^{2}Y^{2}Z^{2}-25X^{3}Z^{3}-96XY^{2}Z^{3}+48Y^{2}Z^{4}+15XZ^{5}-5Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.36.2.z.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{2}w^{3}+\frac{1}{2}w^{2}t-\frac{1}{2}wt^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{13}{80}zw^{8}-\frac{31}{40}zw^{7}t-\frac{71}{80}zw^{6}t^{2}+\frac{67}{40}zw^{5}t^{3}+\frac{43}{16}zw^{4}t^{4}-\frac{51}{20}zw^{3}t^{5}-\frac{39}{20}zw^{2}t^{6}+\frac{12}{5}zwt^{7}-\frac{3}{5}zt^{8}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{10}w^{3}+\frac{1}{10}w^{2}t+\frac{3}{10}wt^{2}-\frac{1}{5}t^{3}$ |
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.
