Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} + x z - 3 y^{2} + z^{2} - w^{2} $ |
| $=$ | $3 x^{2} - 2 x z + 2 y^{2} - 2 z^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 49 x^{4} - 210 x^{2} y^{2} - 56 x^{2} z^{2} + 169 y^{4} + 78 y^{2} z^{2} + 9 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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{7^4}\cdot\frac{1752117229457280xz^{11}-4063963601251008xz^{9}w^{2}+3361247019958464xz^{7}w^{4}-1149047014056032xz^{5}w^{6}+143650302094936xz^{3}w^{8}-8725793922300xzw^{10}+741852195182784z^{12}-1505531655786816z^{10}w^{2}+934428182045904z^{8}w^{4}-104501336462432z^{6}w^{6}-53996894460236z^{4}w^{8}+4449163462380z^{2}w^{10}+318337335875w^{12}}{6756896160xz^{11}+6033966848xz^{9}w^{2}+694641376xz^{7}w^{4}-525223608xz^{5}w^{6}-79570946xz^{3}w^{8}+11138790xzw^{10}+2860892048z^{12}+3454432352z^{10}w^{2}+979138888z^{8}w^{4}-211340584z^{6}w^{6}-96657015z^{4}w^{8}+7083128z^{2}w^{10}+3341637w^{12}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
56.48.1.ce.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 49X^{4}-210X^{2}Y^{2}+169Y^{4}-56X^{2}Z^{2}+78Y^{2}Z^{2}+9Z^{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.
