Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - x z - 2 y w - z^{2} $ |
| $=$ | $3 x^{2} + 2 x z + 2 y^{2} + 2 z^{2} - 2 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 100 x^{4} + 30 x^{2} y^{2} + 40 x^{2} y z - 30 x^{2} z^{2} + y^{4} + 6 y^{3} z + 7 y^{2} z^{2} + \cdots + 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
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{99618750000000xyz^{21}w-7482425000000000xyz^{19}w^{3}+142988908500000000xyz^{17}w^{5}-1080845148800000000xyz^{15}w^{7}+3830887459800000000xyz^{13}w^{9}-6813856941216000000xyz^{11}w^{11}+6115042854432000000xyz^{9}w^{13}-2620123322398720000xyz^{7}w^{15}+462422081387520000xyz^{5}w^{17}-23671044534272000xyz^{3}w^{19}+155729335910400xyzw^{21}+11320312500000xz^{23}-2722889062500000xz^{21}w^{2}+93164971875000000xz^{19}w^{4}-1073431061750000000xz^{17}w^{6}+5436440057250000000xz^{15}w^{8}-13536240326580000000xz^{13}w^{10}+17212818865320000000xz^{11}w^{12}-10999227976656000000xz^{9}w^{14}+3270373206216960000xz^{7}w^{16}-379979999628800000xz^{5}w^{18}+11990542364467200xz^{3}w^{20}-53671118684160xzw^{22}+71692968750000yz^{22}w-7093852812500000yz^{20}w^{3}+174192789875000000yz^{18}w^{5}-1672962998750000000yz^{16}w^{7}+7537612298700000000yz^{14}w^{9}-17257299428616000000yz^{12}w^{11}+20499733362057600000yz^{10}w^{13}-12255270609276160000yz^{8}w^{15}+3342299325982720000yz^{6}w^{17}-331182189498368000yz^{4}w^{19}+6846291743539200yz^{2}w^{21}-9403378237440yw^{23}+6996337890625z^{24}-1764055078125000z^{22}w^{2}+63196726640625000z^{20}w^{4}-764374415562500000z^{18}w^{6}+4099809057093750000z^{16}w^{8}-11021918704320000000z^{14}w^{10}+15727314277220000000z^{12}w^{12}-12156217790668800000z^{10}w^{14}+5048358793757280000z^{8}w^{16}-1067730663960320000z^{6}w^{18}+95597490837964800z^{4}w^{20}-2014782954332160z^{2}w^{22}+2847113842688w^{24}}{w^{4}(475664062500xyz^{17}w-23421325000000xyz^{15}w^{3}+284654606250000xyz^{13}w^{5}-1324585535000000xyz^{11}w^{7}+2782421997000000xyz^{9}w^{9}-2786631180480000xyz^{7}w^{11}+1298183563872000xyz^{5}w^{13}-247206610713600xyz^{3}w^{15}+13038366320640xyzw^{17}+66064453125xz^{19}-10534621093750xz^{17}w^{2}+232265254687500xz^{15}w^{4}-1673747354375000xz^{13}w^{6}+5134290476250000xz^{11}w^{8}-7446253250500000xz^{9}w^{10}+5216079458280000xz^{7}w^{12}-1672827475536000xz^{5}w^{14}+207279970444800xz^{3}w^{16}-6124484597760xzw^{18}+353066406250yz^{18}w-24060132812500yz^{16}w^{3}+391369721875000yz^{14}w^{5}-2401278571250000yz^{12}w^{7}+6655628370100000yz^{10}w^{9}-8955584740440000yz^{8}w^{11}+5856928845840000yz^{6}w^{13}-1724491042502400yz^{4}w^{15}+182596366679040yz^{2}w^{17}-3139403857920yw^{19}+40830078125z^{20}-6894449218750z^{18}w^{2}+160694515625000z^{16}w^{4}-1230443995625000z^{14}w^{6}+4077429313750000z^{12}w^{8}-6624866545900000z^{10}w^{10}+5589825014440000z^{8}w^{12}-2482620600240000z^{6}w^{14}+556204697395200z^{4}w^{16}-52241533148160z^{2}w^{18}+950535005184w^{20})}$ |
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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.
