Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} - x y - x z - 2 y z + y w - z w $ |
| $=$ | $5 x^{2} - 2 x y + 2 x z - 3 y^{2} + 2 y z + z^{2} + w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 13 x^{4} + 48 x^{3} y + 2 x^{3} z + 198 x^{2} y^{2} + 154 x^{2} y z + 42 x^{2} z^{2} - 192 x y^{3} + \cdots + 4 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
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 2\cdot3^3\,\frac{39804413425877466000000xz^{11}-131419532726540843400000xz^{10}w+199890965381524734600000xz^{9}w^{2}-185447673041906753640000xz^{8}w^{3}+116731802403637980840000xz^{7}w^{4}-52217274088961060292000xz^{6}w^{5}+16824226847347110276000xz^{5}w^{6}-3835838766746758917600xz^{4}w^{7}+585374133731325721800xz^{3}w^{8}-50711964647455541380xz^{2}w^{9}+1278601837927489700xzw^{10}+99770635012584428xw^{11}-28771313947097571000000y^{2}z^{10}+94356214691616027000000y^{2}z^{9}w-142771414228914102480000y^{2}z^{8}w^{2}+132070015757824902000000y^{2}z^{7}w^{3}-83158336144180561644000y^{2}z^{6}w^{4}+37388068078659769188000y^{2}z^{5}w^{5}-12195768502018820563200y^{2}z^{4}w^{6}+2852913576714214255200y^{2}z^{3}w^{7}-458938287025075012860y^{2}z^{2}w^{8}+45513744264384434220y^{2}zw^{9}-2121718020800915848y^{2}w^{10}+62035422316497342000000yz^{11}-225725100469604188200000yz^{10}w+380775705542917916760000yz^{9}w^{2}-394907286959150108760000yz^{8}w^{3}+280959465232929108888000yz^{7}w^{4}-144459224302259628900000yz^{6}w^{5}+54918251932499171474400yz^{5}w^{6}-15457797249229352642400yz^{4}w^{7}+3158541183527077826520yz^{3}w^{8}-445680301624721932980yz^{2}w^{9}+38856428427994639756yzw^{10}-1592796073395449948yw^{11}+9134174652895350000000z^{12}-7940473429325115600000z^{11}w-17688434440586357880000z^{10}w^{2}+37313039957346128880000z^{9}w^{3}-29433804420447901314000z^{8}w^{4}+10091617950909212208000z^{7}w^{5}+1402371305534587888800z^{6}w^{6}-3281760831826017108000z^{5}w^{7}+1768472656635778932240z^{4}w^{8}-550209849609811779800z^{3}w^{9}+108090817254251015732z^{2}w^{10}-12423972365640971056zw^{11}+653696708878044523w^{12}}{1911992098189326750000xz^{11}-5183305170569115975000xz^{10}w+6861775021779126375000xz^{9}w^{2}-5816182271484291435000xz^{8}w^{3}+3486746671158518685000xz^{7}w^{4}-1543194462595453438500xz^{6}w^{5}+510732866580834136500xz^{5}w^{6}-124869614284215406800xz^{4}w^{7}+21549074067791760000xz^{3}w^{8}-2329928766078365150xz^{2}w^{9}+99487800527002450xzw^{10}+5601833599051798xw^{11}-1382022872411465925000y^{2}z^{10}+3716016358154086125000y^{2}z^{9}w-4895596965173669775000y^{2}z^{8}w^{2}+4141921752893804670000y^{2}z^{7}w^{3}-2486313309556192126500y^{2}z^{6}w^{4}+1106241158916147685500y^{2}z^{5}w^{5}-370177922266951279950y^{2}z^{4}w^{6}+92396418473044769400y^{2}z^{3}w^{7}-16596371391458603040y^{2}z^{2}w^{8}+1965529955660144010y^{2}zw^{9}-119455077883373603y^{2}w^{10}+2979854584251341850000yz^{11}-9082454241694434975000yz^{10}w+13427097268282857675000yz^{9}w^{2}-12705372710112271815000yz^{8}w^{3}+8547326667351432963000yz^{7}w^{4}-4293454690777986535500yz^{6}w^{5}+1645033448535494009400yz^{5}w^{6}-481817071187458917300yz^{4}w^{7}+106098383040961975380yz^{3}w^{8}-16821173785528725270yz^{2}w^{9}+1740785030999641256yzw^{10}-89236887923639728yw^{11}+438757863665188950000z^{12}-122246586456859350000z^{11}w-848016047067513750000z^{10}w^{2}+1250410303259564400000z^{9}w^{3}-830873320239668060250z^{8}w^{4}+245439478031800518000z^{7}w^{5}+52260712202594195550z^{6}w^{6}-92545532058429313200z^{5}w^{7}+50140840481709285060z^{4}w^{8}-16680796097638317640z^{3}w^{9}+3681948349233432907z^{2}w^{10}-516401531627501126zw^{11}+36674326190447210w^{12}}$ |
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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.
