Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + x z + y^{2} - z^{2} $ |
| $=$ | $5 x^{2} - 4 y^{2} - 2 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2025 x^{4} - 110 x^{2} y^{2} - 270 x^{2} z^{2} + y^{4} + 4 y^{2} z^{2} + 4 z^{4} $ |
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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 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4\cdot3^3}{5^4}\cdot\frac{2136706000000xz^{11}-4082949000000xz^{9}w^{2}+2741419080000xz^{7}w^{4}-736981092000xz^{5}w^{6}+67317172200xz^{3}w^{8}-4191298020xzw^{10}-1027861000000z^{12}+1635542200000z^{10}w^{2}-676786590000z^{8}w^{4}-84555900000z^{6}w^{6}+85573883700z^{4}w^{8}-6504444180z^{2}w^{10}-129730653w^{12}}{854682400xz^{11}+761925600xz^{9}w^{2}+122635296xz^{7}w^{4}-41716296xz^{5}w^{6}-7623882xz^{3}w^{8}+708588xzw^{10}-411144400z^{12}-501471680z^{10}w^{2}-163406808z^{8}w^{4}+11422296z^{6}w^{6}+10126539z^{4}w^{8}-196830z^{2}w^{10}-236196w^{12}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
40.48.1.cg.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 9y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 3w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2025X^{4}-110X^{2}Y^{2}+Y^{4}-270X^{2}Z^{2}+4Y^{2}Z^{2}+4Z^{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.
