Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x^{2} + x u + y t + z v $ |
| $=$ | $x^{2} + x u + 2 y^{2}$ |
| $=$ | $ - x v + y^{2} - 2 y w + y t$ |
| $=$ | $ - x z + y^{2} + 2 y w + y t - z u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{8} z^{2} + 40 x^{6} y^{2} z^{2} - 2 x^{6} z^{4} + 1600 x^{4} y^{6} - 200 x^{4} y^{4} z^{2} + \cdots + 80 y^{8} z^{2} $ |
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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 to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^6}{5^3}\cdot\frac{8408800020836940746308593750xu^{11}-6069472568318926672058593750xu^{10}v+32330625400323036288124218750xu^{9}v^{2}+17096995277582337684123750000xu^{8}v^{3}-77652232510838960096183250000xu^{7}v^{4}+1121355792348961094397044300000xu^{6}v^{5}-7208660520909061283546916620000xu^{5}v^{6}+49486868119362714063078683408000xu^{4}v^{7}-247003777608469924509506606547200xu^{3}v^{8}+287547360945295842035679429524480xu^{2}v^{9}-243586348654027475445107912760832xuv^{10}+1093175840437239189238821100695552xv^{11}-1911764167844040117599484402782208ytv^{10}-24814528532820000000000000zt^{8}v^{3}+29401080556636230000000000000zt^{6}v^{5}-7590019842333653400000000000000zt^{4}v^{7}-792868129017875868750000000000000zt^{2}v^{9}+17300430655907375089677734375zu^{10}v-44187061352083849466867187500zu^{9}v^{2}+351598424283282242683871875000zu^{8}v^{3}+241066564747632242189507500000zu^{7}v^{4}+969349284892638210066493250000zu^{6}v^{5}+2376295191268445406603889200000zu^{5}v^{6}+14647100416090495185097472720000zu^{4}v^{7}-79208099940818404321106543648000zu^{3}v^{8}+358433957381652630010592030163200zu^{2}v^{9}+180756983464327067352497248261120zuv^{10}-2403119215292483486349484402782208zv^{11}-6169301557813862920339843750wtu^{10}+40969712139725282724057812500wtu^{9}v+91833152161510065799273750000wtu^{8}v^{2}+462459946665465290247900500000wtu^{7}v^{3}+1086986391216972571268828300000wtu^{6}v^{4}+915413082495498552237250280000wtu^{5}v^{5}+13394226701451500872859600448000wtu^{4}v^{6}-60311745375819211024714639283200wtu^{3}v^{7}+375522278949770051954501577146880wtu^{2}v^{8}-417683139127503859103127476564992wtuv^{9}+463256743498185807522357798608896wtv^{10}-2757169836980000000000t^{12}+4962905706564000000000000t^{10}v^{2}-2853257205798753000000000000t^{8}v^{4}+298828959856484850000000000000t^{6}v^{6}+125967847111100683162500000000000t^{4}v^{8}+13743778401879201352330078125t^{2}u^{10}+34325200594588541433771093750t^{2}u^{9}v+192567143536477157377705625000t^{2}u^{8}v^{2}+702848379005588469172610250000t^{2}u^{7}v^{3}+509026639402710171151645150000t^{2}u^{6}v^{4}+3183033854040522886328069740000t^{2}u^{5}v^{5}+6105424644049013536462911584000t^{2}u^{4}v^{6}-15942514152825845772729341465600t^{2}u^{3}v^{7}+76092391393973316515565553655040t^{2}u^{2}v^{8}+269322656312859782956909885848064t^{2}uv^{9}-1381409951113698243481823302086656t^{2}v^{10}-2108968217154392261503906250u^{12}-23369903224226301761736328125u^{11}v-64344948674601793612117968750u^{10}v^{2}-391783608643239076789916093750u^{9}v^{3}-411502315880556901417693875000u^{8}v^{4}-1170405331167627671094560200000u^{7}v^{5}-7931939609487313104592233820000u^{6}v^{6}+33048013741170068454058363388000u^{5}v^{7}-186572221895329431337597868579200u^{4}v^{8}+77368999319605938418623410273280u^{3}v^{9}+1150892162420166467613087456437248u^{2}v^{10}+901638485871855513198384288630272uv^{11}+774990023534028814863821100695552v^{12}}{1965463965944824218750xu^{11}-6085540908013183593750xu^{10}v+164246215268186936718750xu^{9}v^{2}-1045449908001227807500000xu^{8}v^{3}+10016792514352345808000000xu^{7}v^{4}-71675019520843306157200000xu^{6}v^{5}+497055968473709269673480000xu^{5}v^{6}-2888379836312281141122432000xu^{4}v^{7}+14130165785181662387704108800xu^{3}v^{8}-16580682194335915155473425920xu^{2}v^{9}+13837028479225812493442518528xuv^{10}-62783612692511177433180049408xv^{11}+110399642357058861500720197632ytv^{10}+17645886956672000000000zt^{8}v^{3}+3176259652200960000000000zt^{6}v^{5}+399017618807745600000000000zt^{4}v^{7}+47221496363989064000000000000zt^{2}v^{9}-1914186166358642578125zu^{10}v+22165806752375976562500zu^{9}v^{2}-196976811713771518750000zu^{8}v^{3}+2083559975608840220000000zu^{7}v^{4}-16499634522488684985500000zu^{6}v^{5}+128229335841088291573200000zu^{5}v^{6}-875806141998179519232880000zu^{4}v^{7}+5252127534003639302027392000zu^{3}v^{8}-21088879154926096509656172800zu^{2}v^{9}-8743706861278328399049876480zuv^{10}+141077621257449186500720197632zv^{11}-724652226874511718750wtu^{10}+22268333180582226562500wtu^{9}v-97835332585705570000000wtu^{8}v^{2}+1640552434563184823000000wtu^{7}v^{3}-11466246447919870768200000wtu^{6}v^{4}+93389258926971105540880000wtu^{5}v^{5}-651328103596282854434592000wtu^{4}v^{6}+3904308086726435695824652800wtu^{3}v^{7}-20549671446668964839440555520wtu^{2}v^{8}+25096829232024421787855111168wtuv^{9}-25018554721991997133639901184wtv^{10}-3529177391334400000000t^{10}v^{2}-379386569568448000000000t^{8}v^{4}-47864468369972800000000000t^{6}v^{6}-5780902853799226400000000000t^{4}v^{8}-362326113437255859375t^{2}u^{10}+9602817657204199218750t^{2}u^{9}v-32774133153549660000000t^{2}u^{8}v^{2}+698129277453543946500000t^{2}u^{7}v^{3}-4328588966439806833100000t^{2}u^{6}v^{4}+36467872519881301782040000t^{2}u^{5}v^{5}-236772507856293426958736000t^{2}u^{4}v^{6}+1371312408546861304526022400t^{2}u^{3}v^{7}-4589604500101449536098364160t^{2}u^{2}v^{8}-15722221790635966313797382656t^{2}uv^{9}+82554844840658550713940148224t^{2}v^{10}+1965463965944824218750u^{12}-4171354741654541015625u^{11}v+143322347426992600781250u^{10}v^{2}-879730755324281972343750u^{9}v^{3}+8178259087236666491125000u^{8}v^{4}-57963536819077954539200000u^{7}v^{5}+392106154933151643862280000u^{6}v^{6}-2197748080701551945845352000u^{5}v^{7}+10268841618667241480797436800u^{4}v^{8}-4513986341511088965324101120u^{3}v^{9}-68465403205047549422718561792u^{2}v^{10}-53165490598640594506917580288uv^{11}-46080828506883829933180049408v^{12}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.7.fc.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 4w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 5X^{8}Z^{2}+40X^{6}Y^{2}Z^{2}-2X^{6}Z^{4}+1600X^{4}Y^{6}-200X^{4}Y^{4}Z^{2}+8X^{4}Y^{2}Z^{4}+160X^{2}Y^{6}Z^{2}-8X^{2}Y^{4}Z^{4}+80Y^{8}Z^{2} $ |
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.
