Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} + x y + x z - 2 y z $ |
| $=$ | $x^{2} - 2 x y - 2 x z + 2 y^{2} + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 26 x^{4} - 4 x^{3} z + 2 x^{2} y^{2} + 6 x^{2} z^{2} - 4 x y^{2} z - 4 x z^{3} + 2 y^{2} z^{2} + z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^6\,\frac{61986931560000000xz^{11}+197291330100000000xz^{9}w^{2}-83826791820720000xz^{7}w^{4}-27388915872240000xz^{5}w^{6}+6471884527308200xz^{3}w^{8}-84793456230600xzw^{10}+74296287957600000y^{2}z^{10}-107640167895000000y^{2}z^{8}w^{2}-23998747550544000y^{2}z^{6}w^{4}+19501433267940000y^{2}z^{4}w^{6}-1504550838998220y^{2}z^{2}w^{8}+9701015407915y^{2}w^{10}-91588010515200000yz^{11}+25252263270000000yz^{9}w^{2}+65584341115488000yz^{7}w^{4}-10694169137080000yz^{5}w^{6}-799184675654560yz^{3}w^{8}-19402030815830yzw^{10}+50120872977600000z^{12}-148184687914200000z^{10}w^{2}-75101035544964000z^{8}w^{4}+26913530122908000z^{6}w^{6}+8860590835417880z^{4}w^{8}-848195265543145z^{2}w^{10}+2989418063996w^{12}}{2295812280000000xz^{11}+3975195868000000xz^{9}w^{2}+2186195443120000xz^{7}w^{4}+513396161928000xz^{5}w^{6}+52916409007800xz^{3}w^{8}+1936281570600xzw^{10}+2751714368800000y^{2}z^{10}+2279453277400000y^{2}z^{8}w^{2}+596329427824000y^{2}z^{6}w^{4}+40995949696000y^{2}z^{4}w^{6}-2343218127380y^{2}z^{2}w^{8}-140471651983y^{2}w^{10}-3392148537600000yz^{11}-3499666034800000yz^{9}w^{2}-1242376222048000yz^{7}w^{4}-175095486592000yz^{5}w^{6}-6990128814240yz^{3}w^{8}+280943303966yzw^{10}+1856328628800000z^{12}+2112410096600000z^{10}w^{2}+755596451844000z^{8}w^{4}+51065667328000z^{6}w^{6}-22718994297480z^{4}w^{8}-3832190539723z^{2}w^{10}-139165071916w^{12}}$ |
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.