Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} + x y + 2 x z - 2 y^{2} + 2 y z + 2 z^{2} $ |
| $=$ | $4 x^{2} - 2 x y - 4 x z - y^{2} - 4 y z - 4 z^{2} - w^{2}$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{3} y + 20 x^{2} z^{2} - 8 x y^{3} + 40 x y z^{2} - y^{4} - 40 y^{2} z^{2} + 225 z^{4} $ |
![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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{10}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^2}{5^2}\cdot\frac{13732536096000000xz^{11}-1698393247200000xz^{9}w^{2}+60184285200000xz^{7}w^{4}-803780656000xz^{5}w^{6}+3807233600xz^{3}w^{8}-4249680xzw^{10}-11910446820000000y^{2}z^{10}+677958118650000y^{2}z^{8}w^{2}-12781674180000y^{2}z^{6}w^{4}+90853320000y^{2}z^{4}w^{6}-200791200y^{2}z^{2}w^{8}+59255y^{2}w^{10}+15490287396000000yz^{11}-1786068964800000yz^{9}w^{2}+61543428480000yz^{7}w^{4}-810802608000yz^{5}w^{6}+3815537200yz^{3}w^{8}-4249680yzw^{10}+19733874426000000z^{12}+1263397338000000z^{10}w^{2}-114610852710000z^{8}w^{4}+2408206316000z^{6}w^{6}-17846125400z^{4}w^{8}+40060360z^{2}w^{10}-11979w^{12}}{1271531120000xz^{11}-169364570000xz^{9}w^{2}+6961532200xz^{7}w^{4}-115615520xz^{5}w^{6}+748784xz^{3}w^{8}-1320xzw^{10}-1102819150000y^{2}z^{10}+73273570625y^{2}z^{8}w^{2}-1687206250y^{2}z^{6}w^{4}+15660875y^{2}z^{4}w^{6}-50050y^{2}z^{2}w^{8}+25y^{2}w^{10}+1434285870000yz^{11}-179032240000yz^{9}w^{2}+7150806200yz^{7}w^{4}-116968320yz^{5}w^{6}+751344yz^{3}w^{8}-1320yzw^{10}+1827210595000z^{12}+99584811000z^{10}w^{2}-11881505525z^{8}w^{4}+311853050z^{6}w^{6}-3051079z^{4}w^{8}+9970z^{2}w^{10}-5w^{12}}$ |
Hi
|
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.
