Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} y + x y z + x z w - 3 y z t - z^{2} t $ |
| $=$ | $x y^{2} + x y z + 2 x y t + x z t + y^{2} z + y z^{2} + y z w + z^{2} w$ |
| $=$ | $2 x^{2} y + x^{2} z - x y^{2} - x y z - 2 x y t - x z t + 2 y^{2} z + 3 y z^{2} - y z w + z^{3} - z^{2} w$ |
| $=$ | $x^{3} + x^{2} z - 2 x^{2} w + x y z + x z^{2} - x z t + y z^{2} - 3 y z w + z^{3} - 3 z^{2} w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{7} + 16 x^{6} y + 17 x^{6} z + 48 x^{5} y^{2} + 96 x^{5} y z - 4 x^{5} z^{2} + 64 x^{4} y^{3} + \cdots + 16 z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ 4x^{8} - 350x^{4} + 156 $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:1)$, $(0:-1:1:0:0)$, $(0:-2:0:-1:1)$, $(0:1:-3:-1:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^{13}\cdot5^6}\cdot\frac{267153456009238726358876541869477024006394431168944440567684521980410508xzt^{12}+130811264511420705058769801974628099790259781132810240xw^{13}+482676619855957478290186251600140381993157133462740992xw^{12}t+11694446190610777820392283150351380261293142175863357440xw^{11}t^{2}-362280279424718153973152657142255454712053848623493558272xw^{10}t^{3}+11921695941219735600277332990567270882603883561398226599936xw^{9}t^{4}-396536989892857064986283289875293794948644375708140935284736xw^{8}t^{5}+374632228724975220601225970049740717743592400614887831625013440xw^{7}t^{6}-10284067540654410810780984782390469415358841656434627454491147264xw^{6}t^{7}+224990387981562824584586249232786860859495113909389297360226843776xw^{5}t^{8}-39813558329608273174344449381692587357359253385268661749194683232092xw^{4}t^{9}+1960404739355730555555830044555097015568346360861570663156880878052952xw^{3}t^{10}-114717218351822606867926517735346359495931257461441921822415921994815782xw^{2}t^{11}-153527019390805550945268910734879678348845736627558609325912465241503869xwt^{12}-4130913841861783799247866782361894002747291629076028084784126724703723xt^{13}+592449916265519663472934734384782161152404206896197164579178631338402968yzwt^{11}+154614856776085791526581751384097999760581110254208518266837890925388106yzt^{12}-275332114311994800911960245550623735419767110232067072yw^{13}+3061284743761196988862173043963188291143051857033033216yw^{12}t-75023280136904324992744677999205328339542395552304140800yw^{11}t^{2}+2102619675818271540891476953149040730614490280998328033792yw^{10}t^{3}-63789785174514326176228116148262502896325239992487943988736yw^{9}t^{4}+2033812421760450425039221094091255689454120101931253091389440yw^{8}t^{5}-826300816087405333711616570693179137943679270042964157538303856yw^{7}t^{6}+34065223625684906869078461783323087074581345581851552879420337584yw^{6}t^{7}-994397431100325250945043945386294471653626917134866093121931901776yw^{5}t^{8}+97959451984458285967145005009077223774676824760250887096643319209552yw^{4}t^{9}-5106075680902106310709944145116375341982851125842779326658184878971324yw^{3}t^{10}+118224995339990283589345299237378707201514216885389734760048736259088669yw^{2}t^{11}+7547737124106476732778785504906956362546356384218313422672383393434827ywt^{12}-113595751217818266414795300229295210009872711873834183621639343027658yt^{13}+833092267708463213121898672585756710535168000000z^{14}+26658952566670822819900757522744214737125376000000z^{12}t^{2}+53317905133341645639801515045488429474250752000000z^{11}wt^{2}-906404387266807975876625755773303301062262784000000z^{11}t^{3}-2132716205333665825592060601819537178970030080000000z^{10}wt^{3}+41428012288606458662125777190344509701492834304000000z^{10}t^{4}+105142908922949725201688587669703182923222482944000000z^{9}wt^{4}-2161721145726203680820112626004282884604022489088000000z^{9}t^{5}-5699470787133688552312222752302531157079508385792000000z^{8}wt^{5}+836218701957898041301664662994520007673160473719746048000000z^{8}t^{6}+3399721732740176455241842887366178392543338052315099136000000z^{7}wt^{6}-94121028543045173828152481702388326150692236814445592473600000z^{7}t^{7}-279891765904149012281416338173775058282734609351144831354880000z^{6}wt^{7}+6260307021120836898803683927159470514972265884956535212144640000z^{6}t^{8}+16483041361750164501381771804823174646564529811832354457837568000z^{5}wt^{8}-2638388496505815516047496551606917653680938283668667746351159232000z^{5}t^{9}-11812641214733165335365290017259407008291945861780064114483117590400z^{4}wt^{9}+274178451937940067703988444247878321967226395973091851460246060720000z^{4}t^{10}+2902951243364425093354102363820358848108488797351885473578150613197760z^{3}wt^{10}-150644084302971493854995503637639431486235670104495212320318606106129520z^{3}t^{11}-836097229720736286397373569055547170544616964041616896z^{2}w^{12}+2891953871491404426076665913444394833965167701391255552z^{2}w^{11}t-83513785252482374135170357150045533459579485156149507072z^{2}w^{10}t^{2}+2289183298997114674120553937666377989480546347343931189248z^{2}w^{9}t^{3}-69067062861404026199388558945484291550919770730005559710208z^{2}w^{8}t^{4}+2195968028068634288720956391608530454633418691593960649936896z^{2}w^{7}t^{5}-2386074489292341909942157812951196003314921953739844071425609664z^{2}w^{6}t^{6}+82346533041720998520780249044138855413687097392976351036402569984z^{2}w^{5}t^{7}-1995892208412169857065452561495641909653687869580279952663319909184z^{2}w^{4}t^{8}+277107970316857680304223072416416768668503494347352779901981281614688z^{2}w^{3}t^{9}-15904677746226250172181731167294439380026395767284097721987871832520800z^{2}w^{2}t^{10}+634924151644455206156553529809657280677318088494152806969437778533275504z^{2}wt^{11}+100161104494410925591002879241070869285761831017394051600991166008285796z^{2}t^{12}+627736678546183204431340422156272775635178234334255104zw^{13}-8031929108144584361359913263644828343309796528182959104zw^{12}t+207977843119586669139786577477178945526607360282402886656zw^{11}t^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|
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.96.3.bh.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{7}+16X^{6}Y+48X^{5}Y^{2}+64X^{4}Y^{3}+32X^{3}Y^{4}+17X^{6}Z+96X^{5}YZ+192X^{4}Y^{2}Z+160X^{3}Y^{3}Z+48X^{2}Y^{4}Z-4X^{5}Z^{2}+92X^{4}YZ^{2}+284X^{3}Y^{2}Z^{2}+144X^{2}Y^{3}Z^{2}+24XY^{4}Z^{2}-63X^{4}Z^{3}-112X^{3}YZ^{3}+196X^{2}Y^{2}Z^{3}+56XY^{3}Z^{3}+4Y^{4}Z^{3}-8X^{3}Z^{4}-212X^{2}YZ^{4}+64XY^{2}Z^{4}+8Y^{3}Z^{4}+100X^{2}Z^{5}-104XYZ^{5}+8Y^{2}Z^{5}+80XZ^{6}-16YZ^{6}+16Z^{7} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.96.3.bh.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{4}{9}y^{6}-\frac{31}{18}y^{5}z-\frac{7}{3}y^{5}t-\frac{139}{12}y^{4}z^{2}-\frac{62}{9}y^{4}zt-4y^{4}t^{2}+\frac{95}{36}y^{3}z^{3}-\frac{263}{12}y^{3}z^{2}t-\frac{86}{9}y^{3}zt^{2}-\frac{20}{9}y^{3}t^{3}+\frac{1193}{18}y^{2}z^{4}-\frac{473}{12}y^{2}z^{3}t-\frac{25}{3}y^{2}z^{2}t^{2}-\frac{10}{3}y^{2}zt^{3}+\frac{758}{9}yz^{5}-\frac{175}{6}yz^{4}t-\frac{19}{6}yz^{3}t^{2}-\frac{5}{3}yz^{2}t^{3}+\frac{278}{9}z^{6}-\frac{64}{9}z^{5}t-\frac{4}{9}z^{4}t^{2}-\frac{5}{18}z^{3}t^{3}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -176y^{21}z^{3}+\frac{17261}{3}y^{20}z^{4}-1024y^{20}z^{3}t-\frac{12957361}{324}y^{19}z^{5}+\frac{275840}{9}y^{19}z^{4}t-1984y^{19}z^{3}t^{2}-\frac{135041485}{1944}y^{18}z^{6}-\frac{4732348}{27}y^{18}z^{5}t+\frac{482396}{9}y^{18}z^{4}t^{2}-1280y^{18}z^{3}t^{3}+\frac{3477740707}{3888}y^{17}z^{7}-\frac{15237358}{27}y^{17}z^{6}t-\frac{748504}{3}y^{17}z^{5}t^{2}+\frac{276560}{9}y^{17}z^{4}t^{3}+\frac{88169082191}{69984}y^{16}z^{8}+\frac{322973963}{81}y^{16}z^{7}t-\frac{96918826}{81}y^{16}z^{6}t^{2}-\frac{10142840}{81}y^{16}z^{5}t^{3}-\frac{796273459963}{104976}y^{15}z^{9}+\frac{15121972037}{1458}y^{15}z^{8}t+\frac{457236134}{81}y^{15}z^{7}t^{2}-\frac{169230520}{243}y^{15}z^{6}t^{3}-\frac{685402863095}{34992}y^{14}z^{10}-\frac{43400885977}{1458}y^{14}z^{9}t+\frac{21022525165}{972}y^{14}z^{8}t^{2}+\frac{657592580}{243}y^{14}z^{7}t^{3}+\frac{97515737837}{34992}y^{13}z^{11}-\frac{1667100118165}{13122}y^{13}z^{10}t-\frac{76485595103}{2187}y^{13}z^{9}t^{2}+\frac{26320176035}{2187}y^{13}z^{8}t^{3}+\frac{475651280963}{6561}y^{12}z^{12}-\frac{276559299737}{4374}y^{12}z^{11}t-\frac{6295778162699}{26244}y^{12}z^{10}t^{2}-\frac{174440334955}{13122}y^{12}z^{9}t^{3}+\frac{738531295405}{3888}y^{11}z^{13}+\frac{5393827282369}{13122}y^{11}z^{12}t-\frac{37501845559}{162}y^{11}z^{11}t^{2}-\frac{803439139415}{6561}y^{11}z^{10}t^{3}+\frac{5221590703999}{11664}y^{10}z^{14}+\frac{14870341280695}{13122}y^{10}z^{13}t+\frac{2161491642469}{2916}y^{10}z^{12}t^{2}-\frac{685737355115}{4374}y^{10}z^{11}t^{3}+\frac{2158973759311}{2916}y^{9}z^{15}+\frac{8516314172035}{4374}y^{9}z^{14}t+\frac{5694712554242}{2187}y^{9}z^{13}t^{2}+\frac{630937363550}{2187}y^{9}z^{12}t^{3}+\frac{75121041817315}{209952}y^{8}z^{16}+\frac{44012283736217}{13122}y^{8}z^{15}t+\frac{11171499281659}{2916}y^{8}z^{14}t^{2}+\frac{5795550026705}{4374}y^{8}z^{13}t^{3}-\frac{3630940882741}{2916}y^{7}z^{17}+\frac{36089120179171}{6561}y^{7}z^{16}t+\frac{13742184389363}{4374}y^{7}z^{15}t^{2}+\frac{5093681593355}{2187}y^{7}z^{14}t^{3}-\frac{27902698713923}{8748}y^{6}z^{18}+\frac{5095659018767}{729}y^{6}z^{17}t+\frac{8090448429929}{6561}y^{6}z^{16}t^{2}+\frac{32999404540945}{13122}y^{6}z^{15}t^{3}-\frac{25328272369798}{6561}y^{5}z^{19}+\frac{41508329701694}{6561}y^{5}z^{18}t-\frac{482379619552}{2187}y^{5}z^{17}t^{2}+\frac{12066472211135}{6561}y^{5}z^{16}t^{3}-\frac{4225525066535}{1458}y^{4}z^{20}+\frac{2904960409184}{729}y^{4}z^{19}t-\frac{1245991734560}{2187}y^{4}z^{18}t^{2}+\frac{681013747040}{729}y^{4}z^{17}t^{3}-\frac{344366530552}{243}y^{3}z^{21}+\frac{415469566280}{243}y^{3}z^{20}t-\frac{81919363072}{243}y^{3}z^{19}t^{2}+\frac{79281931540}{243}y^{3}z^{18}t^{3}-\frac{107349436396}{243}y^{2}z^{22}+\frac{115984681376}{243}y^{2}z^{21}t-\frac{2868328912}{27}y^{2}z^{20}t^{2}+\frac{18212845280}{243}y^{2}z^{19}t^{3}-\frac{720303856}{9}yz^{23}+\frac{704142368}{9}yz^{22}t-\frac{1470612512}{81}yz^{21}t^{2}+\frac{91918640}{9}yz^{20}t^{3}-\frac{19270328}{3}z^{24}+\frac{17144512}{3}z^{23}t-\frac{11961664}{9}z^{22}t^{2}+\frac{1877920}{3}z^{21}t^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{8}{9}y^{6}-\frac{22}{9}y^{5}z-\frac{14}{3}y^{5}t-8y^{4}z^{2}-\frac{124}{9}y^{4}zt-8y^{4}t^{2}-\frac{469}{18}y^{3}z^{3}-\frac{263}{6}y^{3}z^{2}t-\frac{172}{9}y^{3}zt^{2}-\frac{40}{9}y^{3}t^{3}-\frac{703}{18}y^{2}z^{4}-\frac{473}{6}y^{2}z^{3}t-\frac{50}{3}y^{2}z^{2}t^{2}-\frac{20}{3}y^{2}zt^{3}-25yz^{5}-\frac{175}{3}yz^{4}t-\frac{19}{3}yz^{3}t^{2}-\frac{10}{3}yz^{2}t^{3}-\frac{50}{9}z^{6}-\frac{128}{9}z^{5}t-\frac{8}{9}z^{4}t^{2}-\frac{5}{9}z^{3}t^{3}$ |
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.
