Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 8 y^{2} - 2 y z - y w - 3 z^{2} + 2 z w - 2 w^{2} $ |
| $=$ | $35 x^{2} + 3 y^{2} - y z - y w - z^{2} - w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 15 x^{3} z - 280 x^{2} y^{2} + 10 x^{2} z^{2} + 350 x y^{2} z - 10 x z^{3} - 11025 y^{4} + \cdots - 5 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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 40 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^4\cdot5^2\,\frac{34999016960yz^{9}+25251161856yz^{8}w+33462584832yz^{7}w^{2}+24647076608yz^{6}w^{3}+15834409920yz^{5}w^{4}+8400242144yz^{4}w^{5}+3583592096yz^{3}w^{6}+1197205200yz^{2}w^{7}+296910022yzw^{8}+47464587yw^{9}+17551937280z^{10}+4048841728z^{9}w+16718966400z^{8}w^{2}+10265267712z^{7}w^{3}+8629825792z^{6}w^{4}+5465024992z^{5}w^{5}+2846744824z^{4}w^{6}+1261108944z^{3}w^{7}+439789601z^{2}w^{8}+114246794zw^{9}+20954038w^{10}}{104960yz^{9}-1007360yz^{8}w+3473920yz^{7}w^{2}+6424320yz^{6}w^{3}+62036800yz^{5}w^{4}+333310880yz^{4}w^{5}+367786720yz^{3}w^{6}+233385360yz^{2}w^{7}+77998610yzw^{8}+10705525yw^{9}-78592z^{10}+780800z^{9}w-2285440z^{8}w^{2}+4735360z^{7}w^{3}+42020160z^{6}w^{4}+137603648z^{5}w^{5}+95027800z^{4}w^{6}+107171480z^{3}w^{7}+78682255z^{2}w^{8}+30600230zw^{9}+4725322w^{10}}$ |
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.
