Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x^{2} - x y - x z - y^{2} - y z + w^{2} $ |
| $=$ | $x^{2} y + x^{2} z + x y z + x w^{2} - y^{3} - y^{2} z + y z^{2} + y w^{2} + z w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 3 x^{5} y - 3 x^{4} y^{2} + 5 x^{4} z^{2} - 4 x^{3} y^{3} + 5 x^{3} y z^{2} + x^{2} y^{4} + \cdots + y^{2} z^{4} $ |
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This modular curve has 2 rational CM points but no rational cusps or other known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 110 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{687560534690805xyz^{17}-2150079408785697xyz^{15}w^{2}+2726135444501163xyz^{13}w^{4}-1648797116054884xyz^{11}w^{6}+329885833634396xyz^{9}w^{8}+158694916550762xyz^{7}w^{10}-108392975964508xyz^{5}w^{12}+22905444745823xyz^{3}w^{14}-1484310000160xyzw^{16}+162816639423604xz^{18}-351474912976464xz^{16}w^{2}+141305544061490xz^{14}w^{4}+269998473341160xz^{12}w^{6}-317687805415670xz^{10}w^{8}+111226989254604xz^{8}w^{10}+17946116709829xz^{6}w^{12}-20858112698630xz^{4}w^{14}+4333102003822xz^{2}w^{16}-163471360000xw^{18}+312050875671112y^{3}z^{16}-905518986660480y^{3}z^{14}w^{2}+1052336438760643y^{3}z^{12}w^{4}-545422613146037y^{3}z^{10}w^{6}+62608961225427y^{3}z^{8}w^{8}+73711672472754y^{3}z^{6}w^{10}-34324109982693y^{3}z^{4}w^{12}+5323343131149y^{3}z^{2}w^{14}-163322536000y^{3}w^{16}+216251787728720y^{2}z^{17}-518582240909820y^{2}z^{15}w^{2}+447524373284958y^{2}z^{13}w^{4}-34116716180091y^{2}z^{11}w^{6}-104569170260788y^{2}z^{9}w^{8}+63030609321840y^{2}z^{7}w^{10}+1939657398850y^{2}z^{5}w^{12}-5728906256830y^{2}z^{3}w^{14}+941873015820y^{2}zw^{16}+191030944788756yz^{18}-862970835976326yz^{16}w^{2}+1572851998166821yz^{14}w^{4}-1373346198664762yz^{12}w^{6}+611136934857792yz^{10}w^{8}-6359731369335yz^{8}w^{10}-91892837230575yz^{6}w^{12}+38284609597848yz^{4}w^{14}-5402099275309yz^{2}w^{16}+163620184000yw^{18}-295488z^{19}-134602352118212z^{17}w^{2}+241193280070164z^{15}w^{4}-22115618575779z^{13}w^{6}-209855587982010z^{11}w^{8}+186806279635972z^{9}w^{10}-30943671053296z^{7}w^{12}-18929926331119z^{5}w^{14}+9653835265657z^{3}w^{16}-1101067592020zw^{18}}{171xyz^{17}-4041xyz^{15}w^{2}+23333xyz^{13}w^{4}+31764xyz^{11}w^{6}-323141xyz^{9}w^{8}-232894xyz^{7}w^{10}+611794xyz^{5}w^{12}+224809xyz^{3}w^{14}-80738xyzw^{16}+342xz^{18}-7911xz^{16}w^{2}+42796xz^{14}w^{4}+85385xz^{12}w^{6}-629912xz^{10}w^{8}-705196xz^{8}w^{10}+1260798xz^{6}w^{12}+839566xz^{4}w^{14}-397036xz^{2}w^{16}+12900xw^{18}+171y^{3}z^{16}-4383y^{3}z^{14}w^{2}+30902y^{3}z^{12}w^{4}-7909y^{3}z^{10}w^{6}-378872y^{3}z^{8}w^{8}+213771y^{3}z^{6}w^{10}+829133y^{3}z^{4}w^{12}-360086y^{3}z^{2}w^{14}+16261y^{3}w^{16}+513y^{2}z^{17}-13320y^{2}z^{15}w^{2}+96747y^{2}z^{13}w^{4}-48428y^{2}z^{11}w^{6}-1146312y^{2}z^{9}w^{8}+913204y^{2}z^{7}w^{10}+2471200y^{2}z^{5}w^{12}-1500112y^{2}z^{3}w^{14}+140591y^{2}zw^{16}+171yz^{18}-4554yz^{16}w^{2}+35627yz^{14}w^{4}-46038yz^{12}w^{6}-333045yz^{10}w^{8}+602808yz^{8}w^{10}+383574yz^{6}w^{12}-1096361yz^{4}w^{14}+376277yz^{2}w^{16}-10749yw^{18}-171z^{19}+4554z^{17}w^{2}-33062z^{15}w^{4}-3633z^{13}w^{6}+527700z^{11}w^{8}-96364z^{9}w^{10}-2000131z^{7}w^{12}-401809z^{5}w^{14}+988097z^{3}w^{16}-114884zw^{18}}$ |
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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.
