Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - x u + 2 z t $ |
| $=$ | $x t - x u + y u - z t + 2 w t - 2 w u$ |
| $=$ | $4 x^{2} + 2 x y + 3 x w - 2 y^{2} + 3 w^{2} - t u$ |
| $=$ | $x^{2} - 3 x z + 2 x w + 2 y z - 4 z w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 625 x^{8} - 750 x^{7} z - 4500 x^{6} y^{2} - 25 x^{6} z^{2} + 5100 x^{5} y^{2} z + 300 x^{5} z^{3} + \cdots + 36 y^{4} z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ -4x^{8} - 16x^{7} - 28x^{6} - 28x^{5} + 995x^{4} + 2018x^{3} + 1517x^{2} + 506x - 728 $ |
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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:
| $\displaystyle X$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{5}{2}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{5}{2}u$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{675}{332}w^{3}t^{5}-\frac{1125}{664}w^{3}t^{4}u+\frac{9075}{664}w^{3}t^{3}u^{2}-\frac{21675}{1328}w^{3}t^{2}u^{3}+\frac{40125}{5312}w^{3}tu^{4}-\frac{13125}{10624}w^{3}u^{5}+\frac{60}{83}w^{2}t^{6}-\frac{170}{83}w^{2}t^{5}u+\frac{70}{83}w^{2}t^{4}u^{2}+\frac{175}{83}w^{2}t^{3}u^{3}-\frac{725}{332}w^{2}t^{2}u^{4}+\frac{375}{664}w^{2}tu^{5}+\frac{15}{166}wt^{7}+\frac{115}{166}wt^{6}u-\frac{1265}{664}wt^{5}u^{2}+\frac{1435}{664}wt^{4}u^{3}-\frac{4925}{2656}wt^{3}u^{4}+\frac{1425}{1328}wt^{2}u^{5}-\frac{2625}{10624}wtu^{6}-\frac{8}{249}t^{8}-\frac{32}{249}t^{7}u+\frac{140}{249}t^{6}u^{2}-\frac{76}{249}t^{5}u^{3}-\frac{265}{498}t^{4}u^{4}+\frac{145}{249}t^{3}u^{5}-\frac{25}{166}t^{2}u^{6}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{653726375}{379666568}w^{3}t^{26}u^{3}-\frac{43857037175}{2277999408}w^{3}t^{25}u^{4}+\frac{1198710207125}{13667996448}w^{3}t^{24}u^{5}-\frac{5159468689175}{27335992896}w^{3}t^{23}u^{6}+\frac{1233788596675}{13667996448}w^{3}t^{22}u^{7}+\frac{13457147855375}{27335992896}w^{3}t^{21}u^{8}-\frac{21200277325025}{18223995264}w^{3}t^{20}u^{9}+\frac{26814250070875}{36447990528}w^{3}t^{19}u^{10}+\frac{475193573432825}{437375886336}w^{3}t^{18}u^{11}-\frac{2141215310024675}{874751772672}w^{3}t^{17}u^{12}+\frac{2560660988696875}{1749503545344}w^{3}t^{16}u^{13}+\frac{3191949344396375}{3499007090688}w^{3}t^{15}u^{14}-\frac{2432640807726875}{1166335696896}w^{3}t^{14}u^{15}+\frac{999511693601875}{777557131264}w^{3}t^{13}u^{16}+\frac{465641843365625}{13996028362752}w^{3}t^{12}u^{17}-\frac{16518996283371875}{27992056725504}w^{3}t^{11}u^{18}+\frac{100577306015734375}{223936453804032}w^{3}t^{10}u^{19}-\frac{80761211116328125}{447872907608064}w^{3}t^{9}u^{20}+\frac{4250348470703125}{99527312801792}w^{3}t^{8}u^{21}-\frac{1119582146484375}{199054625603584}w^{3}t^{7}u^{22}+\frac{492392578125}{1555114262528}w^{3}t^{6}u^{23}-\frac{608196545}{1708499556}w^{2}t^{27}u^{3}+\frac{46011924145}{10250997336}w^{2}t^{26}u^{4}-\frac{486135757225}{20501994672}w^{2}t^{25}u^{5}+\frac{2613888282715}{41003989344}w^{2}t^{24}u^{6}-\frac{1415483674075}{20501994672}w^{2}t^{23}u^{7}-\frac{1184447264785}{13667996448}w^{2}t^{22}u^{8}+\frac{10694035825045}{27335992896}w^{2}t^{21}u^{9}-\frac{76248072814525}{164015957376}w^{2}t^{20}u^{10}-\frac{51455904321685}{656063829504}w^{2}t^{19}u^{11}+\frac{1139427157912375}{1312127659008}w^{2}t^{18}u^{12}-\frac{2502278212029175}{2624255318016}w^{2}t^{17}u^{13}+\frac{55651910948125}{583167848448}w^{2}t^{16}u^{14}+\frac{151086506943375}{194389282816}w^{2}t^{15}u^{15}-\frac{8542807915409375}{10497021272064}w^{2}t^{14}u^{16}+\frac{5221162397361875}{20994042544128}w^{2}t^{13}u^{17}+\frac{8202677950234375}{41988085088256}w^{2}t^{12}u^{18}-\frac{85504520938496875}{335904680706048}w^{2}t^{11}u^{19}+\frac{30066656576171875}{223936453804032}w^{2}t^{10}u^{20}-\frac{1986849811171875}{49763656400896}w^{2}t^{9}u^{21}+\frac{653307066015625}{99527312801792}w^{2}t^{8}u^{22}-\frac{5810232421875}{12440914100224}w^{2}t^{7}u^{23}-\frac{130745275}{1708499556}wt^{28}u^{3}+\frac{3385917865}{5125498668}wt^{27}u^{4}-\frac{19284464755}{10250997336}wt^{26}u^{5}+\frac{2540158325}{10250997336}wt^{25}u^{6}+\frac{777307457305}{82007978688}wt^{24}u^{7}-\frac{18492376925}{1138999704}wt^{23}u^{8}-\frac{3148342385}{379666568}wt^{22}u^{9}+\frac{4172102454935}{82007978688}wt^{21}u^{10}-\frac{22431419621225}{656063829504}wt^{20}u^{11}-\frac{37106385423935}{656063829504}wt^{19}u^{12}+\frac{126242875099025}{1312127659008}wt^{18}u^{13}-\frac{371290779275}{54671985792}wt^{17}u^{14}-\frac{318598989469625}{3499007090688}wt^{16}u^{15}+\frac{89376615159625}{1312127659008}wt^{15}u^{16}+\frac{94274952791875}{5248510636032}wt^{14}u^{17}-\frac{1040863302509375}{20994042544128}wt^{13}u^{18}+\frac{7296729522221875}{335904680706048}wt^{12}u^{19}+\frac{591347549159375}{111968226902016}wt^{11}u^{20}-\frac{2033671628828125}{223936453804032}wt^{10}u^{21}+\frac{97273541640625}{24881828200448}wt^{9}u^{22}-\frac{156596516015625}{199054625603584}wt^{8}u^{23}+\frac{98478515625}{1555114262528}wt^{7}u^{24}+\frac{121639309}{7688248002}t^{29}u^{3}-\frac{4827070801}{30752992008}t^{28}u^{4}+\frac{33771735097}{61505984016}t^{27}u^{5}-\frac{47605735259}{123011968032}t^{26}u^{6}-\frac{634218832807}{246023936064}t^{25}u^{7}+\frac{875914889879}{123011968032}t^{24}u^{8}-\frac{647332860023}{246023936064}t^{23}u^{9}-\frac{8264163130799}{492047872128}t^{22}u^{10}+\frac{1683207422483}{61505984016}t^{21}u^{11}+\frac{4165429942513}{3936382977024}t^{20}u^{12}-\frac{341598607274185}{7872765954048}t^{19}u^{13}+\frac{600547243416515}{15745531908096}t^{18}u^{14}+\frac{431328667534075}{31491063816192}t^{17}u^{15}-\frac{1360415111755525}{31491063816192}t^{16}u^{16}+\frac{1314187743281125}{62982127632384}t^{15}u^{17}+\frac{16393388060375}{1517641629696}t^{14}u^{18}-\frac{8020785984835625}{503857021059072}t^{13}u^{19}+\frac{10373823435494375}{2015428084236288}t^{12}u^{20}+\frac{2029240906196875}{1343618722824192}t^{11}u^{21}-\frac{492610485765625}{298581938405376}t^{10}u^{22}+\frac{289885741796875}{597163876810752}t^{9}u^{23}-\frac{321696484375}{6220457050112}t^{8}u^{24}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{675}{332}w^{3}t^{5}+\frac{1125}{664}w^{3}t^{4}u-\frac{9075}{664}w^{3}t^{3}u^{2}+\frac{21675}{1328}w^{3}t^{2}u^{3}-\frac{40125}{5312}w^{3}tu^{4}+\frac{13125}{10624}w^{3}u^{5}-\frac{60}{83}w^{2}t^{6}+\frac{170}{83}w^{2}t^{5}u-\frac{70}{83}w^{2}t^{4}u^{2}-\frac{175}{83}w^{2}t^{3}u^{3}+\frac{725}{332}w^{2}t^{2}u^{4}-\frac{375}{664}w^{2}tu^{5}-\frac{15}{166}wt^{7}-\frac{115}{166}wt^{6}u+\frac{1265}{664}wt^{5}u^{2}-\frac{1435}{664}wt^{4}u^{3}+\frac{4925}{2656}wt^{3}u^{4}-\frac{1425}{1328}wt^{2}u^{5}+\frac{2625}{10624}wtu^{6}+\frac{8}{249}t^{8}-\frac{19}{498}t^{7}u-\frac{47}{996}t^{6}u^{2}-\frac{7}{996}t^{5}u^{3}+\frac{145}{1992}t^{4}u^{4}+\frac{545}{7968}t^{3}u^{5}-\frac{425}{5312}t^{2}u^{6}$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^6\cdot5^2}\cdot\frac{153857568608409375xw^{9}-144451816041498750xw^{7}u^{2}+8145909819120375xw^{5}u^{4}+11633894099323425xw^{3}u^{6}+12765173218012380xwu^{8}-37697358789731250yw^{9}+87503119219867500yw^{7}u^{2}-47580375420012375yw^{5}u^{4}+5483323953096750yw^{3}u^{6}-47628654778080ywt^{8}+1149912968017680ywt^{7}u-10636582642277040ywt^{6}u^{2}+46827867812673960ywt^{5}u^{3}-103704984198189450ywt^{4}u^{4}+120628210163714745ywt^{3}u^{5}-67438028856400125ywt^{2}u^{6}+9256353123465480ywtu^{7}+4657400464121115ywu^{8}+59142346058396400z^{2}u^{8}-214543916924146875zw^{9}+151537023178143750zw^{7}u^{2}+39112323850282500zw^{5}u^{4}-28325247230104875zw^{3}u^{6}+1511654400000zwu^{8}+12565596937387500w^{10}-23061343040724375w^{8}u^{2}+9368315984875500w^{6}u^{4}-572218866825750w^{4}u^{6}+13168623856802310w^{2}u^{8}+16445931349824t^{10}-407752535809440t^{9}u+3928891485583728t^{8}u^{2}-18702807113894736t^{7}u^{3}+47809068167910220t^{6}u^{4}-71173910410446002t^{5}u^{5}+62960071080611393t^{4}u^{6}-30568624379919880t^{3}u^{7}+5688489752520342t^{2}u^{8}+1320325590685900tu^{9}-3942991592031760u^{10}}{911250xw^{5}u^{4}+33975xw^{3}u^{6}-2488440xwu^{8}+1093500yw^{5}u^{4}-382050yw^{3}u^{6}-4199040000ywt^{8}+24494400000ywt^{7}u-61236000000ywt^{6}u^{2}+85633200000ywt^{5}u^{3}-73256402160ywt^{4}u^{4}+39281740920ywt^{3}u^{5}-12909616920ywt^{2}u^{6}+2380767750ywtu^{7}-188939025ywu^{8}-10173600z^{2}u^{8}-2004750zw^{5}u^{4}+7875zw^{3}u^{6}-1093500w^{6}u^{4}+442800w^{4}u^{6}-2498115w^{2}u^{8}+1440944640t^{10}-9378201600t^{9}u+27293760000t^{8}u^{2}-46787395200t^{7}u^{3}+52127106656t^{6}u^{4}-39071095904t^{5}u^{5}+19610256936t^{4}u^{6}-6317697836t^{3}u^{7}+1178759518t^{2}u^{8}-96597825tu^{9}+678240u^{10}}$ |
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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.
