Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x z - x w + x t + y t $ |
| $=$ | $4 x y + x z + x w + x t + 2 y^{2} + y z + y w + z^{2} + 2 z w - w^{2} - t^{2}$ |
| $=$ | $7 x^{2} - x w - y w - 2 z w + w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 98 x^{6} - 49 x^{5} z - 16 x^{4} y^{2} + 49 x^{4} z^{2} - 33 x^{3} y^{2} z + 2 x^{2} y^{4} + \cdots - 4 y^{2} z^{4} $ |
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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.
| Canonical model |
|---|
| $(0:0:-1:-2:1)$, $(0:0:-1:0:1)$, $(0:0:1:0:1)$, $(0:0:1:2:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^8\cdot7^6}\cdot\frac{872034098749636608000xw^{11}+1090942699601452859392xw^{10}t-22582454329613580107776xw^{9}t^{2}-339541896047599562096640xw^{8}t^{3}+61564950168840857804800xw^{7}t^{4}+2486836841387241699829760xw^{6}t^{5}+264042294726726649165312xw^{5}t^{6}-2593210050496223996985728xw^{4}t^{7}+1975468035935519517191520xw^{3}t^{8}+2387229165391424420368232xw^{2}t^{9}+2015679070935260170087158xwt^{10}+1630570309380709810176xt^{11}+29299256245489867685888y^{2}w^{9}t+51269371584216965709824y^{2}w^{8}t^{2}-570493557789054255333376y^{2}w^{7}t^{3}-998153149270945216503808y^{2}w^{6}t^{4}+2478762507445454985259008y^{2}w^{5}t^{5}+4334147456709445815154176y^{2}w^{4}t^{6}-1932785316347288295227776y^{2}w^{3}t^{7}-3346976353010241064129184y^{2}w^{2}t^{8}+2348044268062998263025512y^{2}wt^{9}+3976536292286776597277430y^{2}t^{10}+93301981509058560yz^{11}-93301981509058560yz^{10}t-338997199482912768yz^{9}t^{2}+478950171746500608yz^{8}t^{3}+62297733053598400512yz^{7}t^{4}-63832550649422413824yz^{6}t^{5}-974335228363651022848yz^{5}t^{6}+1077703716161024425984yz^{4}t^{7}+24874137087096160059392yz^{3}t^{8}-27031275012003031089152yz^{2}t^{9}+1488947320960630962353531yzt^{10}+871940796768127549440yw^{11}+20927107155534814904320yw^{10}t-37213969369392104210432yw^{9}t^{2}-407425648969442917646336yw^{8}t^{3}+346311116247914864320512yw^{7}t^{4}+1769135106674042671183872yw^{6}t^{5}-968842556845519369764352yw^{5}t^{6}-1365094208386862445304960yw^{4}t^{7}+2890582073262729917533728yw^{3}t^{8}+1584792867891950033156280yw^{2}t^{9}-639889736713302796857913ywt^{10}-1486892156658897904301435yt^{11}+47687679437963264z^{12}+227034821672042496z^{10}t^{2}-46650990754529280z^{9}t^{3}+28985815588814192640z^{8}t^{4}+1055867424077512704z^{7}t^{5}-70392457466021806080z^{6}t^{6}-49659202141683843072z^{5}t^{7}+3600472020758654418944z^{4}t^{8}+1395095027620318674944z^{3}t^{9}+2824490674421192720384z^{2}w^{10}-14646852845395740196864z^{2}w^{9}t-109378077228274760220672z^{2}w^{8}t^{2}+285081076334413143916544z^{2}w^{7}t^{3}+77203465533261117345792z^{2}w^{6}t^{4}-1236434184107852219298816z^{2}w^{5}t^{5}+904965674144063653651712z^{2}w^{4}t^{6}+940219997478859809448128z^{2}w^{3}t^{7}-2300446965482177097176752z^{2}w^{2}t^{8}-1089817359297671544203188z^{2}wt^{9}+2017508617244846393554299z^{2}t^{10}-1080609080884937621504zw^{11}+56501394160286444027904zw^{10}t-52158121707699722846208zw^{9}t^{2}-1099998858517946202439680zw^{8}t^{3}+1036386154091715890835456zw^{7}t^{4}+4776534562474228499850240zw^{6}t^{5}-604810517088594740551936zw^{5}t^{6}-3693949696638478294686144zw^{4}t^{7}+1398350199360560571757296zw^{3}t^{8}+4424281966557391640254756zw^{2}t^{9}+1778089737295839929179022zwt^{10}-1346445041911957815296zt^{11}-69189701973803270144w^{12}-20926501139540798341120w^{11}t+46815819140896935772160w^{10}t^{2}+392764250839942589612032w^{9}t^{3}-501564109375144380407808w^{8}t^{4}-1483986059238594564782080w^{7}t^{5}+290244104716791787867648w^{6}t^{6}+130093162565461962057856w^{5}t^{7}-1445667032783319018039584w^{4}t^{8}-676011200135901475461112w^{3}t^{9}+265283467012445028998025w^{2}t^{10}-1179891578194337037707188wt^{11}-2021339719064459068667259t^{12}}{807403520xw^{10}t-8165785600xw^{9}t^{2}-139709251584xw^{8}t^{3}+374794649600xw^{7}t^{4}+1008851400704xw^{6}t^{5}+4605702932992xw^{5}t^{6}+16006919187840xw^{4}t^{7}-12317753184352xw^{3}t^{8}-9021189116072xw^{2}t^{9}-69574694182xwt^{10}-220200960y^{2}w^{9}t-385351680y^{2}w^{8}t^{2}-92233924608y^{2}w^{7}t^{3}-161409368064y^{2}w^{6}t^{4}+3132675438592y^{2}w^{5}t^{5}+5559849949696y^{2}w^{4}t^{6}+3384461585792y^{2}w^{3}t^{7}+4423474935712y^{2}w^{2}t^{8}-8089798420648y^{2}wt^{9}-10756748542246y^{2}t^{10}-3097866240yz^{7}t^{4}+3097866240yz^{6}t^{5}+48567658496yz^{5}t^{6}-53214457856yz^{4}t^{7}-345239982080yz^{3}t^{8}+452167998464yz^{2}t^{9}-5268924935827yzt^{10}-157286400yw^{10}t-8055685120yw^{9}t^{2}-65881374720yw^{8}t^{3}+424009478144yw^{7}t^{4}+2256212510720yw^{6}t^{5}+2845134620160yw^{5}t^{6}+1856104777856yw^{4}t^{7}-12531839559456yw^{3}t^{8}-3157202345080yw^{2}t^{9}+8868977377729ywt^{10}+5166643718803yt^{11}-1583353856z^{8}t^{4}+5266372608z^{6}t^{6}+1548933120z^{5}t^{7}+254610184192z^{4}t^{8}-53713558528z^{3}t^{9}-29360128z^{2}w^{10}+110100480z^{2}w^{9}t+20524564480z^{2}w^{8}t^{2}+46116962304z^{2}w^{7}t^{3}-68293750784z^{2}w^{6}t^{4}-1614639143936z^{2}w^{5}t^{5}-10951185722624z^{2}w^{4}t^{6}-694944057024z^{2}w^{3}t^{7}+25441545003056z^{2}w^{2}t^{8}+1799944387668z^{2}wt^{9}-11671904667283z^{2}t^{10}+29360128zw^{11}-424673280zw^{10}t-33039319040zw^{9}t^{2}-177879711744zw^{8}t^{3}+682969419776zw^{7}t^{4}+6116000357376zw^{6}t^{5}+11275454695680zw^{5}t^{6}+5113885081536zw^{4}t^{7}+1208592906640zw^{3}t^{8}-13013004963140zw^{2}t^{9}-16159723740286zwt^{10}+52164625408zt^{11}-8388608w^{12}+157286400w^{11}t+11964776448w^{10}t^{2}+65991475200w^{9}t^{3}-318732029952w^{8}t^{4}-2217618937856w^{7}t^{5}-3040637978112w^{6}t^{6}-3507903564928w^{5}t^{7}-3551898676704w^{4}t^{8}+3244248237496w^{3}t^{9}-3829881086353w^{2}t^{10}+4245537680468wt^{11}+11413611464339t^{12}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
28.96.5.k.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 98X^{6}-16X^{4}Y^{2}+2X^{2}Y^{4}-49X^{5}Z-33X^{3}Y^{2}Z-XY^{4}Z+49X^{4}Z^{2}+29X^{2}Y^{2}Z^{2}+Y^{4}Z^{2}+8XY^{2}Z^{3}-4Y^{2}Z^{4} $ |
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.
