Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 4 x^{2} + 2 x y - 2 x z - 2 x w - y^{2} + 2 y z + 2 y w - z^{2} - 2 z w + w^{2} $ |
| $=$ | $2 x^{2} + x y - x z - 3 y^{2} - 4 y z - 3 z^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{3} y - 5 x^{3} z + 6 x^{2} y^{2} - 5 x^{2} y z + 10 x^{2} z^{2} + 4 x y^{3} + 5 x y^{2} z + \cdots + 25 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{5}\cdot\frac{25344000000xyz^{10}+233696000000xyz^{9}w+730015200000xyz^{8}w^{2}+1096316800000xyz^{7}w^{3}+902756600000xyz^{6}w^{4}+431962920000xyz^{5}w^{5}+122664360000xyz^{4}w^{6}+20544976000xyz^{3}w^{7}+1953486300xyz^{2}w^{8}+95707100xyzw^{9}+1842875xyw^{10}+20736000000xz^{11}+128806400000xz^{10}w+224288800000xz^{9}w^{2}+58042560000xz^{8}w^{3}-209663000000xz^{7}w^{4}-246176072000xz^{6}w^{5}-121608008000xz^{5}w^{6}-31941156000xz^{4}w^{7}-4674602700xz^{3}w^{8}-373841500xz^{2}w^{9}-14930435xzw^{10}-227859xw^{11}+17088000000yz^{11}+186841600000yz^{10}w+694475200000yz^{9}w^{2}+1247783040000yz^{8}w^{3}+1238265600000yz^{7}w^{4}+720973812000yz^{6}w^{5}+252592004000yz^{5}w^{6}+53380734000yz^{4}w^{7}+6676644400yz^{3}w^{8}+470567950yz^{2}w^{9}+16773310yzw^{10}+227859yw^{11}+14240000000z^{12}+116262400000z^{11}w+313152000000z^{10}w^{2}+369298560000z^{9}w^{3}+202132160000z^{8}w^{4}+63687308000z^{7}w^{5}+41995488000z^{6}w^{6}+33791374000z^{5}w^{7}+14011904200z^{4}w^{8}+2978858050z^{3}w^{9}+330175420z^{2}w^{10}+17894211zw^{11}+368607w^{12}}{w^{4}(1176000xyz^{6}+4148800xyz^{5}w+4508000xyz^{4}w^{2}+2029760xyz^{3}w^{3}+406780xyz^{2}w^{4}+34700xyzw^{5}+995xyw^{6}+840000xz^{7}+1201600xz^{6}w-641120xz^{5}w^{2}-1364640xz^{4}w^{3}-576684xz^{3}w^{4}-93212xz^{2}w^{5}-5963xzw^{6}-123xw^{7}+832000yz^{7}+3936800yz^{6}w+5789920yz^{5}w^{2}+3559120yz^{4}w^{3}+1000304yz^{3}w^{4}+128462yz^{2}w^{5}+6958yzw^{6}+123yw^{7}+624000z^{8}+1810400z^{7}w+1182080z^{6}w^{2}+40080z^{5}w^{3}+122216z^{4}w^{4}+195330z^{3}w^{5}+60156z^{2}w^{6}+6267zw^{7}+199w^{8})}$ |
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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.
