Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ - x^{2} z + x^{2} t + y^{3} - 2 y^{2} t + y z t + y t^{2} - z t^{2} $ |
| $=$ | $x^{3} - x^{2} y - x^{2} z - 3 x^{2} w + x y^{2} - x y t + x z t - y^{2} w + y w t - z w t$ |
| $=$ | $x^{2} y - x^{2} z - x y w - x y t + x z w + x z t - y z w - y w^{2} - y w t - z^{2} w - 2 z w^{2} + \cdots + w t^{2}$ |
| $=$ | $x^{3} - x^{2} y - x^{2} z + 2 x^{2} w + x y^{2} - 2 x y t + 2 x t^{2} + y^{2} t - y z w + y z t + \cdots - z t^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{7} - 16 x^{6} y + 16 x^{6} z + 24 x^{5} y^{2} + 28 x^{5} y z - 64 x^{5} z^{2} - 16 x^{4} y^{3} + \cdots + 25 y^{2} z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 10x^{6} - 13x^{4} + 250x^{2} + 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 |
|---|
| $(0:2:-2:0:1)$, $(-1:1:1:-1:1)$, $(0:0:0:1:0)$, $(1:1:1:0:1)$ |
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 \frac{2}{5}\cdot\frac{260607500xzw^{9}+74239500xzw^{8}t-591428400xzw^{7}t^{2}-17854800xzw^{6}t^{3}+447702880xzw^{5}t^{4}-134902560xzw^{4}t^{5}-113710960xzw^{3}t^{6}+69616880xzw^{2}t^{7}-11664000xzwt^{8}+88000xzt^{9}+1017855375xw^{10}+35455875xw^{9}t-2378069750xw^{8}t^{2}+581397650xw^{7}t^{3}+1836390300xw^{6}t^{4}-1020425480xw^{5}t^{5}-362669120xw^{4}t^{6}+408699520xw^{3}t^{7}-100792120xw^{2}t^{8}+7863120xwt^{9}+420000xt^{10}-383078275yzw^{9}+242508900yzw^{8}t+621815450yzw^{7}t^{2}-543251620yzw^{6}t^{3}-154959720yzw^{5}t^{4}+295636184yzw^{4}t^{5}-88683080yzw^{3}t^{6}+2085608yzw^{2}t^{7}+1989872yzwt^{8}+188800yzt^{9}-20480625yw^{10}-1151257725yw^{9}t+770525350yw^{8}t^{2}+1812657050yw^{7}t^{3}-1689963980yw^{6}t^{4}-408662680yw^{5}t^{5}+893611056yw^{4}t^{6}-292780640yw^{3}t^{7}+13448632yw^{2}t^{8}+6170128ywt^{9}-452800yt^{10}-326860675z^{2}w^{9}+267801800z^{2}w^{8}t+505693850z^{2}w^{7}t^{2}-552453940z^{2}w^{6}t^{3}-61955880z^{2}w^{5}t^{4}+271975128z^{2}w^{4}t^{5}-115050488z^{2}w^{3}t^{6}+14693720z^{2}w^{2}t^{7}+139376z^{2}wt^{8}+49600z^{2}t^{9}-579072000zw^{10}-415231250zw^{9}t+1662463500zw^{8}t^{2}+402792500zw^{7}t^{3}-1644495800zw^{6}t^{4}+346666800zw^{5}t^{5}+535781040zw^{4}t^{6}-309989264zw^{3}t^{7}+50055552zw^{2}t^{8}-59744zwt^{9}-584000zt^{10}-1250w^{11}-1117185250w^{10}t+1945888050w^{9}t^{2}+1124620700w^{8}t^{3}-3503658100w^{7}t^{4}+1222838840w^{6}t^{5}+1323383320w^{5}t^{6}-1173927008w^{4}t^{7}+290695632w^{3}t^{8}-7066032w^{2}t^{9}-5842752wt^{10}+294400t^{11}}{w^{2}(17000xzw^{7}+25000xzw^{6}t-16240xzw^{5}t^{2}-46720xzw^{4}t^{3}+13280xzw^{3}t^{4}+12860xzw^{2}t^{5}-4650xzwt^{6}+250xzt^{7}+67800xw^{8}+64500xw^{7}t-88550xw^{6}t^{2}-105960xw^{5}t^{3}+49460xw^{4}t^{4}+36815xw^{3}t^{5}-22440xw^{2}t^{6}+2840xwt^{7}+25xt^{8}-26480yzw^{7}-7220yzw^{6}t+39010yzw^{5}t^{2}+5318yzw^{4}t^{3}-17290yzw^{3}t^{4}+2641yzw^{2}t^{5}+709yzwt^{6}-65yzt^{7}-2000yw^{8}-77420yw^{7}t-16930yw^{6}t^{2}+103240yw^{5}t^{3}+18912yw^{4}t^{4}-50525yw^{3}t^{5}+8714yw^{2}t^{6}+1716ywt^{7}-185yt^{8}-22760z^{2}w^{7}-2940z^{2}w^{6}t+36570z^{2}w^{5}t^{2}+806z^{2}w^{4}t^{3}-17186z^{2}w^{3}t^{4}+4585z^{2}w^{2}t^{5}+147z^{2}wt^{6}-55z^{2}t^{7}-40400zw^{8}-71200zw^{7}t+58300zw^{6}t^{2}+114140zw^{5}t^{3}-34320zw^{4}t^{4}-41628zw^{3}t^{5}+15514zw^{2}t^{6}-518zwt^{7}-100zt^{8}-76800w^{8}t+48860w^{7}t^{2}+151740w^{6}t^{3}-75180w^{5}t^{4}-85016w^{4}t^{5}+60874w^{3}t^{6}-7044w^{2}t^{7}-1594wt^{8}+130t^{9})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
20.72.3.o.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4X^{7}-16X^{6}Y+24X^{5}Y^{2}-16X^{4}Y^{3}+4X^{3}Y^{4}+16X^{6}Z+28X^{5}YZ-82X^{4}Y^{2}Z+76X^{3}Y^{3}Z-13X^{2}Y^{4}Z-64X^{5}Z^{2}+70X^{4}YZ^{2}+89X^{3}Y^{2}Z^{2}-130X^{2}Y^{3}Z^{2}+14XY^{4}Z^{2}+60X^{4}Z^{3}-200X^{3}YZ^{3}-10X^{2}Y^{2}Z^{3}+95XY^{3}Z^{3}-5Y^{4}Z^{3}-40X^{3}Z^{4}+145X^{2}YZ^{4}-45XY^{2}Z^{4}-25Y^{3}Z^{4}+50X^{2}Z^{5}-25XYZ^{5}+25Y^{2}Z^{5}-25XZ^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
20.72.3.o.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{127}{8100}x^{10}+\frac{7057}{97200}x^{9}w-\frac{1}{900}x^{9}t-\frac{5111}{24300}x^{8}w^{2}+\frac{781}{32400}x^{8}wt+\frac{29}{1350}x^{8}t^{2}+\frac{45349}{194400}x^{7}w^{3}-\frac{7}{270}x^{7}w^{2}t-\frac{2381}{16200}x^{7}wt^{2}-\frac{19}{4050}x^{7}t^{3}+\frac{22853}{38880}x^{6}w^{4}-\frac{4}{25}x^{6}w^{3}t+\frac{10279}{32400}x^{6}w^{2}t^{2}+\frac{89}{4050}x^{6}wt^{3}-\frac{4271}{2160}x^{5}w^{5}+\frac{10441}{32400}x^{5}w^{4}t+\frac{1169}{64800}x^{5}w^{3}t^{2}-\frac{1177}{64800}x^{5}w^{2}t^{3}+\frac{5221}{2592}x^{4}w^{6}+\frac{1249}{12960}x^{4}w^{5}t-\frac{3871}{3240}x^{4}w^{4}t^{2}-\frac{1253}{16200}x^{4}w^{3}t^{3}-\frac{4855}{7776}x^{3}w^{7}-\frac{3317}{4320}x^{3}w^{6}t+\frac{3547}{1620}x^{3}w^{5}t^{2}+\frac{4643}{21600}x^{3}w^{4}t^{3}-\frac{497}{2592}x^{2}w^{8}+\frac{2119}{2592}x^{2}w^{7}t-\frac{2003}{1080}x^{2}w^{6}t^{2}-\frac{497}{2160}x^{2}w^{5}t^{3}+\frac{145}{1296}xw^{9}-\frac{319}{864}xw^{8}t+\frac{1015}{1296}xw^{7}t^{2}+\frac{151}{1296}xw^{6}t^{3}+\frac{55}{864}w^{9}t-\frac{115}{864}w^{8}t^{2}-\frac{5}{216}w^{7}t^{3}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{9763}{1133740800}x^{40}-\frac{367825139}{2295825120000}x^{39}w-\frac{3767}{94478400}x^{39}t+\frac{44203226203537}{37192366944000000}x^{38}w^{2}+\frac{369282527}{382637520000}x^{38}wt+\frac{3233}{62985600}x^{38}t^{2}-\frac{14135639925454229}{5578855041600000000}x^{37}w^{3}-\frac{3861989827937}{344373768000000}x^{37}w^{2}t-\frac{342058873}{255091680000}x^{37}wt^{2}-\frac{3941}{141717600}x^{37}t^{3}-\frac{4230948930980143}{142818689064960000}x^{36}w^{4}+\frac{57344680729583329}{697356880200000000}x^{36}w^{3}t+\frac{105747237398101}{6198727824000000}x^{36}w^{2}t^{2}+\frac{864321217}{1147912560000}x^{36}wt^{3}+\frac{171588470923948247}{495898225920000000}x^{35}w^{5}-\frac{623162337350148593}{1487694677760000000}x^{35}w^{4}t-\frac{195245617484612113}{1394713760400000000}x^{35}w^{3}t^{2}-\frac{23261047514017}{2324522934000000}x^{35}w^{2}t^{3}-\frac{419554128043141649}{206624260800000000}x^{34}w^{6}+\frac{301919408739945809}{198359290368000000}x^{34}w^{5}t+\frac{12296839427528688137}{14876946777600000000}x^{34}w^{4}t^{2}+\frac{120530799445180931}{1394713760400000000}x^{34}w^{3}t^{3}+\frac{18278061450674482451}{2231542016640000000}x^{33}w^{7}-\frac{9353887601604858649}{2479491129600000000}x^{33}w^{6}t-\frac{110219474158336108927}{29753893555200000000}x^{33}w^{5}t^{2}-\frac{1207331777920156667}{2231542016640000000}x^{33}w^{4}t^{3}-\frac{1455689917329113058541}{59507787110400000000}x^{32}w^{8}+\frac{421652655620146602347}{89261680665600000000}x^{32}w^{7}t+\frac{576635320385442968831}{44630840332800000000}x^{32}w^{6}t^{2}+\frac{115687683030616124911}{44630840332800000000}x^{32}w^{5}t^{3}+\frac{180290108879683727263}{3305988172800000000}x^{31}w^{9}+\frac{667203983555561643311}{89261680665600000000}x^{31}w^{8}t-\frac{3139844671447956893933}{89261680665600000000}x^{31}w^{7}t^{2}-\frac{58109495073571217581}{5950778711040000000}x^{31}w^{6}t^{3}-\frac{15258790521577348980193}{178523361331200000000}x^{30}w^{10}-\frac{5301146874761068949051}{89261680665600000000}x^{30}w^{9}t+\frac{3247671042803719603271}{44630840332800000000}x^{30}w^{8}t^{2}+\frac{867618693534397280239}{29753893555200000000}x^{30}w^{7}t^{3}+\frac{1506783372180359674273}{22315420166400000000}x^{29}w^{11}+\frac{1568603394981656206493}{8926168066560000000}x^{29}w^{10}t-\frac{6124706609475103138021}{59507787110400000000}x^{29}w^{9}t^{2}-\frac{506602687457043301561}{7438473388800000000}x^{29}w^{8}t^{3}+\frac{119680055399985949921241}{1428186890649600000000}x^{28}w^{12}-\frac{11249322461879606252101}{35704672266240000000}x^{28}w^{11}t+\frac{906622576241910007421}{17852336133120000000}x^{28}w^{10}t^{2}+\frac{10500698345851210212179}{89261680665600000000}x^{28}w^{9}t^{3}-\frac{151113883426074008613149}{357046722662400000000}x^{27}w^{13}+\frac{20076339434372760215281}{71409344532480000000}x^{27}w^{12}t+\frac{18000638060937451855261}{89261680665600000000}x^{27}w^{11}t^{2}-\frac{21478758694766492059837}{178523361331200000000}x^{27}w^{10}t^{3}+\frac{607420171111999801486907}{714093445324800000000}x^{26}w^{14}+\frac{182261984410798328406881}{714093445324800000000}x^{26}w^{13}t-\frac{504220837853723472955037}{714093445324800000000}x^{26}w^{12}t^{2}-\frac{440418966893976403729}{8926168066560000000}x^{26}w^{11}t^{3}-\frac{365984086776280819200337}{357046722662400000000}x^{25}w^{15}-\frac{51658510653495541857467}{35704672266240000000}x^{25}w^{14}t+\frac{2837746263391214888891}{2285099025039360000}x^{25}w^{13}t^{2}+\frac{10314677525346428630527}{19835929036800000000}x^{25}w^{12}t^{3}+\frac{229118088070520478907061}{476062296883200000000}x^{24}w^{16}+\frac{261442485096728926173589}{95212459376640000000}x^{24}w^{15}t-\frac{171994488968533201348909}{142818689064960000000}x^{24}w^{14}t^{2}-\frac{290576048016866224166237}{238031148441600000000}x^{24}w^{13}t^{3}+\frac{1410699004552833468713933}{1428186890649600000000}x^{23}w^{17}-\frac{3908109885656852514357089}{1428186890649600000000}x^{23}w^{16}t-\frac{131109601875250048572251}{1428186890649600000000}x^{23}w^{15}t^{2}+\frac{786521751397819334052131}{476062296883200000000}x^{23}w^{14}t^{3}-\frac{412733653941158499116579}{142818689064960000000}x^{22}w^{18}+\frac{17986622342998983294851}{714093445324800000000}x^{22}w^{17}t+\frac{750832159549799386372123}{285637378129920000000}x^{22}w^{16}t^{2}-\frac{1389753164068398223679041}{1428186890649600000000}x^{22}w^{15}t^{3}+\frac{474382489093252760928419}{119015574220800000000}x^{21}w^{19}+\frac{7163459219383733698102501}{1428186890649600000000}x^{21}w^{18}t-\frac{717953608494640793332753}{142818689064960000000}x^{21}w^{17}t^{2}-\frac{93851960701800356523739}{71409344532480000000}x^{21}w^{16}t^{3}-\frac{533086530011761124758301}{178523361331200000000}x^{20}w^{20}-\frac{4436266736866249973198047}{476062296883200000000}x^{20}w^{19}t+\frac{398500987610199026220281}{79343716147200000000}x^{20}w^{18}t^{2}+\frac{6318799420471923898415369}{1428186890649600000000}x^{20}w^{17}t^{3}-\frac{39384789712217245419587}{142818689064960000000}x^{19}w^{21}+\frac{167288002625416427332031}{19042491875328000000}x^{19}w^{20}t-\frac{12549286598873773539823}{9521245937664000000}x^{19}w^{19}t^{2}-\frac{586641700782887144403791}{95212459376640000000}x^{19}w^{18}t^{3}+\frac{601818914878293949311629}{142818689064960000000}x^{18}w^{22}-\frac{22545488562871057296527}{11425495125196800000}x^{18}w^{21}t-\frac{21990173364603196609361}{4760622968832000000}x^{18}w^{20}t^{2}+\frac{695196726164261880845897}{158687432294400000000}x^{18}w^{19}t^{3}-\frac{120946743999048494083843}{19042491875328000000}x^{17}w^{23}-\frac{144887926924721901399239}{19042491875328000000}x^{17}w^{22}t+\frac{1278566072477767743106859}{142818689064960000000}x^{17}w^{21}t^{2}+\frac{79932644218493099932363}{95212459376640000000}x^{17}w^{20}t^{3}+\frac{300395735745190162813811}{57127475625984000000}x^{16}w^{24}+\frac{259163912745552847458521}{19042491875328000000}x^{16}w^{23}t-\frac{53322388605546471081517}{6347497291776000000}x^{16}w^{22}t^{2}-\frac{1873300831525306980699371}{285637378129920000000}x^{16}w^{21}t^{3}-\frac{3382820934382801827913}{1904249187532800000}x^{15}w^{25}-\frac{17185625410294376616383}{1428186890649600000}x^{15}w^{24}t+\frac{30194730180376649349209}{9521245937664000000}x^{15}w^{23}t^{2}+\frac{516524234618498600763457}{57127475625984000000}x^{15}w^{22}t^{3}-\frac{267928469410518160253}{152339935002624000}x^{14}w^{26}+\frac{1301336769294364742899}{285637378129920000}x^{14}w^{25}t+\frac{17774514456653492122649}{5712747562598400000}x^{14}w^{24}t^{2}-\frac{97771821543336845573011}{14281868906496000000}x^{14}w^{23}t^{3}+\frac{7967914971043571465381}{2285099025039360000}x^{13}w^{27}+\frac{1895557956676919646929}{571274756259840000}x^{13}w^{26}t-\frac{37252413244456754561117}{5712747562598400000}x^{13}w^{25}t^{2}+\frac{1334316110959304475959}{714093445324800000}x^{13}w^{24}t^{3}-\frac{2443228458759787223461}{761699675013120000}x^{12}w^{28}-\frac{8144065379363768719031}{1142549512519680000}x^{12}w^{27}t+\frac{13398488785264813112929}{2285099025039360000}x^{12}w^{26}t^{2}+\frac{14650714337326740128471}{5712747562598400000}x^{12}w^{25}t^{3}+\frac{885507753527335473197}{457019805007872000}x^{11}w^{29}+\frac{976657740657936945553}{152339935002624000}x^{11}w^{28}t-\frac{61042884295593348281}{21158324305920000}x^{11}w^{27}t^{2}-\frac{2522545464960354256913}{571274756259840000}x^{11}w^{26}t^{3}-\frac{5693960382247011233}{7616996750131200}x^{10}w^{30}-\frac{110085809319613766069}{30467987000524800}x^{10}w^{29}t+\frac{15959532248060085877}{152339935002624000}x^{10}w^{28}t^{2}+\frac{2966290325300530637831}{761699675013120000}x^{10}w^{27}t^{3}+\frac{110778092492847115}{1218719480020992}x^{9}w^{31}+\frac{60692905877835169}{50779978334208}x^{9}w^{30}t+\frac{20114028673677345259}{16926659444736000}x^{9}w^{29}t^{2}-\frac{366470514504355948513}{152339935002624000}x^{9}w^{28}t^{3}+\frac{1134913636169274961}{10155995666841600}x^{8}w^{32}-\frac{13999303937566303}{406239826673664}x^{8}w^{31}t-\frac{12223483996288414253}{10155995666841600}x^{8}w^{30}t^{2}+\frac{2353093583177700011}{2115832430592000}x^{8}w^{29}t^{3}-\frac{69568720237757827}{677066377789440}x^{7}w^{33}-\frac{2299284789112003}{10579162152960}x^{7}w^{32}t+\frac{308437377275790211}{423166486118400}x^{7}w^{31}t^{2}-\frac{1318344342038104423}{3385331888947200}x^{7}w^{30}t^{3}+\frac{1447976641428187}{28211099074560}x^{6}w^{34}+\frac{48496025228696327}{338533188894720}x^{6}w^{33}t-\frac{212363619445830163}{677066377789440}x^{6}w^{32}t^{2}+\frac{172137249848798359}{1692665944473600}x^{6}w^{31}t^{3}-\frac{2405976658317317}{135413275557888}x^{5}w^{35}-\frac{2436563244039529}{45137758519296}x^{5}w^{34}t+\frac{832018898519843}{8358844170240}x^{5}w^{33}t^{2}-\frac{6411493190239157}{338533188894720}x^{5}w^{32}t^{3}+\frac{49913617888633}{11284439629824}x^{4}w^{36}+\frac{308046090573101}{22568879259648}x^{4}w^{35}t-\frac{1047328032154765}{45137758519296}x^{4}w^{34}t^{2}+\frac{102092034382955}{45137758519296}x^{4}w^{33}t^{3}-\frac{35086499619785}{45137758519296}x^{3}w^{37}-\frac{35410792989295}{15045919506432}x^{3}w^{36}t+\frac{19339004818475}{5015306502144}x^{3}w^{35}t^{2}-\frac{2644506561625}{22568879259648}x^{3}w^{34}t^{3}+\frac{685239204425}{7522959753216}x^{2}w^{38}+\frac{1999847383675}{7522959753216}x^{2}w^{37}t-\frac{6473911515875}{15045919506432}x^{2}w^{36}t^{2}-\frac{41408157625}{5015306502144}x^{2}w^{35}t^{3}-\frac{294487625}{46438023168}xw^{39}-\frac{88790159125}{5015306502144}xw^{38}t+\frac{143992725625}{5015306502144}xw^{37}t^{2}+\frac{4048820875}{2507653251072}xw^{36}t^{3}+\frac{9150625}{46438023168}w^{40}+\frac{9150625}{17414258688}w^{39}t-\frac{239723125}{278628139008}w^{38}t^{2}-\frac{40339375}{557256278016}w^{37}t^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{421}{24300}x^{10}+\frac{1561}{19440}x^{9}w+\frac{1}{1350}x^{9}t-\frac{527}{10800}x^{8}w^{2}-\frac{781}{48600}x^{8}wt-\frac{29}{2025}x^{8}t^{2}-\frac{77983}{194400}x^{7}w^{3}+\frac{7}{405}x^{7}w^{2}t+\frac{2381}{24300}x^{7}wt^{2}+\frac{19}{6075}x^{7}t^{3}+\frac{6761}{7200}x^{6}w^{4}+\frac{8}{75}x^{6}w^{3}t-\frac{10279}{48600}x^{6}w^{2}t^{2}-\frac{89}{6075}x^{6}wt^{3}-\frac{14587}{48600}x^{5}w^{5}-\frac{10441}{48600}x^{5}w^{4}t-\frac{1169}{97200}x^{5}w^{3}t^{2}+\frac{1177}{97200}x^{5}w^{2}t^{3}-\frac{20129}{12960}x^{4}w^{6}-\frac{1249}{19440}x^{4}w^{5}t+\frac{3871}{4860}x^{4}w^{4}t^{2}+\frac{1253}{24300}x^{4}w^{3}t^{3}+\frac{8251}{3240}x^{3}w^{7}+\frac{3317}{6480}x^{3}w^{6}t-\frac{3547}{2430}x^{3}w^{5}t^{2}-\frac{4643}{32400}x^{3}w^{4}t^{3}-\frac{6727}{3888}x^{2}w^{8}-\frac{2119}{3888}x^{2}w^{7}t+\frac{2003}{1620}x^{2}w^{6}t^{2}+\frac{497}{3240}x^{2}w^{5}t^{3}+\frac{79}{144}xw^{9}+\frac{319}{1296}xw^{8}t-\frac{1015}{1944}xw^{7}t^{2}-\frac{151}{1944}xw^{6}t^{3}-\frac{55}{864}w^{10}-\frac{55}{1296}w^{9}t+\frac{115}{1296}w^{8}t^{2}+\frac{5}{324}w^{7}t^{3}$ |
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.
