Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - 8 x y + x z + 6 y^{2} + y z - z^{2} $ |
| $=$ | $3 x^{2} + 6 x y - 2 x z + 13 y^{2} - 2 y z + 2 z^{2} - 3 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 4 x^{3} y + 18 x^{2} y^{2} - 45 x^{2} z^{2} - 68 x y^{3} + 240 x y z^{2} + 169 y^{4} + \cdots + 225 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 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2\cdot3^2\,\frac{2052698391605255488403842674432000000000xz^{17}-21180579187626525313189265686272000000000xz^{15}w^{2}-19046371115119607633242261839014400000000xz^{13}w^{4}+171836720184942284378322807070745728000000xz^{11}w^{6}-193564505572949519418229295892881712000000xz^{9}w^{8}+74245744883681461162805025605760227520000xz^{7}w^{10}-106576929145974526652763559146274730640000xz^{5}w^{12}+152407608507984704314302078290586118620000xz^{3}w^{14}-60202383027303337937774820239652970355610xzw^{16}+22526699173751936289625277768448000000000y^{2}z^{16}+3680432138503351232215567411968000000000y^{2}z^{14}w^{2}-515479894425041668880264112006336000000000y^{2}z^{12}w^{4}+562895421904106202265750379126545792000000y^{2}z^{10}w^{6}+242915243850567408397334894264829648000000y^{2}z^{8}w^{8}+760866190151123316767517472688924143680000y^{2}z^{6}w^{10}-1886470619157575357153729777552969702640000y^{2}z^{4}w^{12}+554871146853931928242812289275098470740000y^{2}z^{2}w^{14}+254766692666632346090050453785021869378910y^{2}w^{16}-754162666373992506106258804992000000000yz^{17}+30389165647519359284198580745267200000000yz^{15}w^{2}+7126739687897995680778015141209600000000yz^{13}w^{4}-400574310924655785646490052244433792000000yz^{11}w^{6}+865497297554339758868180374674961872000000yz^{9}w^{8}-942733075216660109251767359997184653120000yz^{7}w^{10}+1003489517992346361655508614503316910832000yz^{5}w^{12}-895036306820615845912883053643399293860000yz^{3}w^{14}+332864682215243536036163090019451061076390yzw^{16}+235830455490550089306277093632000000000z^{18}+11447181432001239403465228467686400000000z^{16}w^{2}-42066800541302034668643039503040000000000z^{14}w^{4}+84724315650922873217795137114742912000000z^{12}w^{6}-131056395587060136269777147463285609600000z^{10}w^{8}+187541909588267249908585602011075550720000z^{8}w^{10}-253863269258193508926072655795215792816000z^{6}w^{12}+213361926210687109563857345060364135948000z^{4}w^{14}-66688514794645458938331480976046591616390z^{2}w^{16}-3619479684675859219272890944528009575723w^{18}}{21382274912554744670873361192000000000xz^{17}+7963732863533544957656201208000000000xz^{15}w^{2}-289396680493193225458850732984700000000xz^{13}w^{4}+531708280200090215548054090576812000000xz^{11}w^{6}-363503234862796193797941633386819250000xz^{9}w^{8}+119301717176038638630986357168940870000xz^{7}w^{10}-19548280651503353313040967299047688125xz^{5}w^{12}+1416767347557935161489038500660788875xz^{3}w^{14}-25194272745919542147387764400125205xzw^{16}+234653116393249336350263310088000000000y^{2}z^{16}-978861620192957170382781132792000000000y^{2}z^{14}w^{2}+788629413733595665174260091082100000000y^{2}z^{12}w^{4}+670913715942020511282364110811548000000y^{2}z^{10}w^{6}-1088976813002550131079029083328234250000y^{2}z^{8}w^{8}+548593811015519644407413494488073530000y^{2}z^{6}w^{10}-135241593756691289241975477876320855625y^{2}z^{4}w^{12}+16806170718794497408359762325266079875y^{2}z^{2}w^{14}-845974580126950521152480781697828545y^{2}w^{16}-7855861108062421938606862552000000000yz^{17}-99793065312289027858148739796800000000yz^{15}w^{2}+423664315389367375724235788736900000000yz^{13}w^{4}-571750377753008598714487889665668000000yz^{11}w^{6}+354956798661363811705174055464956750000yz^{9}w^{8}-114422005791680447021769463867386570000yz^{7}w^{10}+19060143931489873703575547392757143875yz^{5}w^{12}-1527910865153849618569320081603170625yz^{3}w^{14}+40919091179510795352472275853295295yzw^{16}+2456567244693230096940386392000000000z^{18}-91604271718764481696834006161600000000z^{16}w^{2}+310085484144402495442543487789100000000z^{14}w^{4}-298252219850856278727733164225672000000z^{12}w^{6}-11222916934504223678239549032429150000z^{10}w^{8}+141693259278070634132101836435757470000z^{8}w^{10}-79798774882399206473456030546484870375z^{6}w^{12}+20228424770126592364464801082446698250z^{4}w^{14}-2521584479111403497590312222691772270z^{2}w^{16}+126066240293555229718864177613655081w^{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.
