Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x^{2} + y z $ |
| $=$ | $x^{2} + 2 y^{2} - y z - y w + z w - w^{2}$ |
| $=$ | $x^{2} + x y - x z + x w + 2 x u + y^{2} - y w + y t + z t - w t$ |
| $=$ | $x w - 2 x t + 2 y w + 2 y u - z w + w^{2} + w u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 392 x^{6} - 448 x^{5} y - 140 x^{5} z + 236 x^{4} y^{2} - 248 x^{4} y z - 114 x^{4} z^{2} + 20 x^{3} y^{3} + \cdots + z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 5x^{8} + 6x^{7} + 28x^{6} - 42x^{5} + 17x^{4} + 42x^{3} + 28x^{2} - 6x + 5 $ |
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This modular curve has no $\Q_p$ points for $p=23$, and therefore no rational points.
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{2^2}{5^4}\cdot\frac{36169916542883611021347750xt^{11}+1414998199394147197626385950xt^{10}u-4267657853680120609414526100xt^{9}u^{2}-15691160103583337044740513000xt^{8}u^{3}+30398152100507209797515936100xt^{7}u^{4}+30862895337559385714435402700xt^{6}u^{5}-42715172336665134023029150650xt^{5}u^{6}-14767926749769811824286366950xt^{4}u^{7}+14025082200612887098500068700xt^{3}u^{8}+1521276902730578995842224250xt^{2}u^{9}-799904403664256574286346475xtu^{10}-16753209984432339184462275xu^{11}+53064292224502857689417070yt^{11}-526608184867654162252002210yt^{10}u-3056291820829003712338946070yt^{9}u^{2}+9313775455947637017815306640yt^{8}u^{3}+18887153332134143961129633120yt^{7}u^{4}-32016022899819141284658326700yt^{6}u^{5}-24895140769550492884627475130yt^{5}u^{6}+28051305421726121558270649750yt^{4}u^{7}+6983142221170234294736632020yt^{3}u^{8}-5636697127245275318728631910yt^{2}u^{9}-182058820673273289676946835ytu^{10}+114589233572500117478429775yu^{11}+79093069720000000000z^{12}+14236752549600000000z^{8}u^{4}-5694701019840000000z^{7}u^{5}-854205152976000000z^{6}u^{6}+1879251336547200000z^{5}u^{7}-199314535694400000z^{4}u^{8}-522773553621312000z^{3}u^{9}+323345123906515200z^{2}u^{10}+42321872643441764453447310zt^{11}-315554825345884194420231480zt^{10}u-1383430951513805888357715600zt^{9}u^{2}+4088852396768539294976361960zt^{8}u^{3}+4730358610310773635161230320zt^{7}u^{4}-8257801340974026154849361280zt^{6}u^{5}-3523140767233243356547922820zt^{5}u^{6}+3643177107204497275962168600zt^{4}u^{7}+532205760745004535722203920zt^{3}u^{8}-277743074006621442827976750zt^{2}u^{9}-6701283993772935673784910ztu^{10}+11744165391464698189200zu^{11}+5427480078867463807910013w^{2}t^{10}+177434208965193888701481642w^{2}t^{9}u-591243089670377292758303484w^{2}t^{8}u^{2}-1056341924500273148233594224w^{2}t^{7}u^{3}+3319184011332591921562520274w^{2}t^{6}u^{4}-123819925888746397933056w^{2}t^{5}u^{5}-3318825212109652022317236690w^{2}t^{4}u^{6}+1056314196047825920621665648w^{2}t^{3}u^{7}+590894244710308829547186300w^{2}t^{2}u^{8}-177303793565059938108974250w^{2}tu^{9}-5416301363536876424722173w^{2}u^{10}-36883005943150352989287297wt^{11}+478521067693924638713174022wt^{10}u+260716446599410882317955626wt^{9}u^{2}-3768030878371584548651661024wt^{8}u^{3}+3742088159401163294871230874wt^{7}u^{4}-7638732174344843973297456wt^{6}u^{5}-8053405215820986561205441230wt^{5}u^{6}+6884329741205173328776068468wt^{4}u^{7}+2637545042658871633294425780wt^{3}u^{8}-2126559538140578235205352820wt^{2}u^{9}-60672811463063883065671083wtu^{10}+49989724355164647117856140wu^{11}-5434517643497542567675638t^{12}-184668192273770610905751192t^{11}u+339928403820834906120416364t^{10}u^{2}+1483478679838832300288220264t^{9}u^{3}-992120231545142871821938704t^{8}u^{4}-1560103903600031184387596604t^{7}u^{5}-603550059408182160555483510t^{6}u^{6}-245933791322798578193275158t^{5}u^{7}+981178561413634888185656310t^{4}u^{8}+182185759688135272500233070t^{3}u^{9}-119050719434792786061204597t^{2}u^{10}-3350641996886467836892455tu^{11}-7258738731213897192825u^{12}}{348266525750xt^{11}+6685678713100xt^{10}u-47788493236725xt^{9}u^{2}+94047009476275xt^{8}u^{3}-12861503350700xt^{7}u^{4}-180342958066250xt^{6}u^{5}+232872257667525xt^{5}u^{6}-89838303876975xt^{4}u^{7}-24462184649100xt^{3}u^{8}+25998467522800xt^{2}u^{9}-4497194806750xtu^{10}-161041918950xu^{11}+510409764810yt^{11}-5145229060460yt^{10}u+3607563521455yt^{9}u^{2}+48458755345165yt^{8}u^{3}-126973535496560yt^{7}u^{4}+83833105567050yt^{6}u^{5}+82499932026855yt^{5}u^{6}-155099691740645yt^{4}u^{7}+78994074604640yt^{3}u^{8}-5793128412160yt^{2}u^{9}-5996987201160ytu^{10}+1104528388590yu^{11}+406894016980zt^{11}-3479108028230zt^{10}u+6246216376150zt^{9}u^{2}+7950215988880zt^{8}u^{3}-36962155699210zt^{7}u^{4}+40757067961750zt^{6}u^{5}-10439058099330zt^{5}u^{6}-11194791430000zt^{4}u^{7}+8171087155030zt^{3}u^{8}-1391700377860zt^{2}u^{9}-64416767580ztu^{10}+52422889904w^{2}t^{10}+665198383682w^{2}t^{9}u-5436008816217w^{2}t^{8}u^{2}+14047742135392w^{2}t^{7}u^{3}-14708203888357w^{2}t^{6}u^{4}+14708203888357w^{2}t^{4}u^{6}-14047742135392w^{2}t^{3}u^{7}+5436008816217w^{2}t^{2}u^{8}-665198383682w^{2}tu^{9}-52422889904w^{2}u^{10}-354471127076wt^{11}+4004999801612wt^{10}u-14024208975487wt^{9}u^{2}+22034340181852wt^{8}u^{3}-9009042669707wt^{7}u^{4}-30184406220570wt^{6}u^{5}+65904329949437wt^{5}u^{6}-60826826245112wt^{4}u^{7}+25283340700627wt^{3}u^{8}-793008509982wt^{2}u^{9}-2516830745294wtu^{10}+481783859700wu^{11}-52422889904t^{12}-734851688832t^{11}u+4403299453657t^{10}u^{2}-8042704641537t^{9}u^{3}+5239764922282t^{8}u^{4}+477977058720t^{7}u^{5}-2691623103197t^{6}u^{6}+4019859273857t^{5}u^{7}-4870346257012t^{4}u^{8}+2787372437382t^{3}u^{9}-505144770026t^{2}u^{10}-32208383790tu^{11}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.96.3.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2u$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 392X^{6}-448X^{5}Y+236X^{4}Y^{2}+20X^{3}Y^{3}-25X^{2}Y^{4}-140X^{5}Z-248X^{4}YZ+126X^{3}Y^{2}Z+10X^{2}Y^{3}Z-114X^{4}Z^{2}+112X^{3}YZ^{2}-59X^{2}Y^{2}Z^{2}-5XY^{3}Z^{2}+176X^{3}Z^{3}+12X^{2}YZ^{3}-4XY^{2}Z^{3}-34X^{2}Z^{4}-3XYZ^{4}+Y^{2}Z^{4}-6XZ^{5}+Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.96.3.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{86}w^{7}-\frac{101}{1505}w^{6}t+\frac{307}{3010}w^{6}u+\frac{227}{1505}w^{5}t^{2}-\frac{389}{3010}w^{5}tu-\frac{2}{301}w^{5}u^{2}-\frac{394}{2107}w^{4}t^{3}+\frac{205}{2107}w^{4}t^{2}u+\frac{1}{86}w^{4}tu^{2}-\frac{15}{602}w^{4}u^{3}+\frac{32}{301}w^{3}t^{4}-\frac{157}{2107}w^{3}t^{3}u-\frac{6}{2107}w^{3}t^{2}u^{2}+\frac{5}{301}w^{3}tu^{3}-\frac{22}{2107}w^{2}t^{5}+\frac{40}{2107}w^{2}t^{4}u+\frac{4}{301}w^{2}t^{3}u^{2}-\frac{10}{2107}w^{2}t^{2}u^{3}+\frac{44}{10535}wt^{6}-\frac{64}{10535}wt^{5}u-\frac{4}{2107}wt^{4}u^{2}-\frac{8}{10535}t^{7}+\frac{8}{10535}t^{6}u$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{56271}{168560000}w^{28}+\frac{242603729}{50736560000}w^{27}t+\frac{48577}{58996000}w^{27}u-\frac{123727267129}{3817926140000}w^{26}t^{2}-\frac{96175319}{8878898000}w^{26}tu-\frac{79529}{165188800}w^{26}u^{2}+\frac{91317973135583}{656683296080000}w^{25}t^{3}+\frac{126227579671}{1908963070000}w^{25}t^{2}u+\frac{44475439}{7103118400}w^{25}tu^{2}+\frac{153}{16518880}w^{25}u^{3}-\frac{488554908519227}{1149195768140000}w^{24}t^{4}-\frac{72158821618777}{287298942035000}w^{24}t^{3}u-\frac{201193874453}{5345096596000}w^{24}t^{2}u^{2}-\frac{59541}{710311840}w^{24}tu^{3}+\frac{4525554107017369}{4596783072560000}w^{23}t^{5}+\frac{769928466302449}{1149195768140000}w^{23}t^{4}u+\frac{12912362646569}{91935661451200}w^{23}t^{3}u^{2}+\frac{33577553}{133627414900}w^{23}t^{2}u^{3}-\frac{57525482374133479}{32177481507920000}w^{22}t^{6}-\frac{1344625974524193}{1005546297122500}w^{22}t^{5}u-\frac{591365891813089}{1608874075396000}w^{22}t^{4}u^{2}-\frac{2532175767}{45967830725600}w^{22}t^{3}u^{3}+\frac{10433437843149459}{4022185188490000}w^{21}t^{7}+\frac{8317680021868211}{4022185188490000}w^{21}t^{6}u+\frac{2313446990252399}{3217748150792000}w^{21}t^{5}u^{2}-\frac{5117350529}{3217748150792}w^{21}t^{4}u^{3}-\frac{85257798026153089}{28155296319430000}w^{20}t^{8}-\frac{70987295394972801}{28155296319430000}w^{20}t^{7}u-\frac{4898937959817833}{4504847411108800}w^{20}t^{6}u^{2}+\frac{319879606987}{64354963015840}w^{20}t^{5}u^{3}+\frac{79617553269702747}{28155296319430000}w^{19}t^{9}+\frac{136862196890056651}{56310592638860000}w^{19}t^{8}u+\frac{3638693444210047}{2815529631943000}w^{19}t^{7}u^{2}-\frac{3516688397961}{450484741110880}w^{19}t^{6}u^{3}-\frac{405693254275972107}{197087074236010000}w^{18}t^{10}-\frac{179732142664256899}{98543537118005000}w^{18}t^{9}u-\frac{9517948188523881}{7883482969440400}w^{18}t^{8}u^{2}+\frac{1748009828487}{281552963194300}w^{18}t^{7}u^{3}+\frac{13309665344298086}{12317942139750625}w^{17}t^{11}+\frac{99403222020608717}{98543537118005000}w^{17}t^{10}u+\frac{8564441938294891}{9854353711800500}w^{17}t^{9}u^{2}+\frac{4299975319977}{3941741484720200}w^{17}t^{8}u^{3}-\frac{7137161060114651}{24635884279501250}w^{16}t^{12}-\frac{584617115661016}{1759706019964375}w^{16}t^{11}u-\frac{4452540305207213}{9854353711800500}w^{16}t^{10}u^{2}-\frac{400373720163}{39417414847202}w^{16}t^{9}u^{3}-\frac{1565505216395156}{12317942139750625}w^{15}t^{13}-\frac{139118612395829}{3519412039928750}w^{15}t^{12}u+\frac{64257212484848}{492717685590025}w^{15}t^{11}u^{2}+\frac{3012825815253}{197087074236010}w^{15}t^{10}u^{3}+\frac{64834958644516}{286463770691875}w^{14}t^{14}+\frac{1831412361657652}{12317942139750625}w^{14}t^{13}u+\frac{69522309978776}{2463588427950125}w^{14}t^{12}u^{2}-\frac{199117732288}{14077648159715}w^{14}t^{11}u^{3}-\frac{2107809374695144}{12317942139750625}w^{13}t^{15}-\frac{1539281587787556}{12317942139750625}w^{13}t^{14}u-\frac{31067358215816}{492717685590025}w^{13}t^{13}u^{2}+\frac{4595412299292}{492717685590025}w^{13}t^{12}u^{3}+\frac{160125938768076}{1759706019964375}w^{12}t^{16}+\frac{881786472749248}{12317942139750625}w^{12}t^{15}u+\frac{111763829276728}{2463588427950125}w^{12}t^{14}u^{2}-\frac{2250394241072}{492717685590025}w^{12}t^{13}u^{3}-\frac{66364628216288}{1759706019964375}w^{11}t^{17}-\frac{396305539047024}{12317942139750625}w^{11}t^{16}u-\frac{54924703640384}{2463588427950125}w^{11}t^{15}u^{2}+\frac{34012537040}{19708707423601}w^{11}t^{14}u^{3}+\frac{156337485373728}{12317942139750625}w^{10}t^{18}+\frac{146492997760192}{12317942139750625}w^{10}t^{17}u+\frac{598615489184}{70388240798575}w^{10}t^{16}u^{2}-\frac{10253820672}{19708707423601}w^{10}t^{15}u^{3}-\frac{43426713049344}{12317942139750625}w^{9}t^{19}-\frac{45535774326976}{12317942139750625}w^{9}t^{18}u-\frac{6477443479488}{2463588427950125}w^{9}t^{17}u^{2}+\frac{57842112}{458342033107}w^{9}t^{16}u^{3}+\frac{9993080012032}{12317942139750625}w^{8}t^{20}+\frac{12030251184128}{12317942139750625}w^{8}t^{19}u+\frac{1536691904}{2291710165535}w^{8}t^{18}u^{2}-\frac{11858484096}{492717685590025}w^{8}t^{17}u^{3}-\frac{1894803213056}{12317942139750625}w^{7}t^{21}-\frac{2704504125824}{12317942139750625}w^{7}t^{20}u-\frac{350490932224}{2463588427950125}w^{7}t^{19}u^{2}+\frac{1779993216}{492717685590025}w^{7}t^{18}u^{3}+\frac{289238375168}{12317942139750625}w^{6}t^{22}+\frac{516260326912}{12317942139750625}w^{6}t^{21}u+\frac{1251845888}{50277314856125}w^{6}t^{20}u^{2}-\frac{8030208}{19708707423601}w^{6}t^{19}u^{3}-\frac{4842053632}{1759706019964375}w^{5}t^{23}-\frac{82832805376}{12317942139750625}w^{5}t^{22}u-\frac{349727744}{98543537118005}w^{5}t^{21}u^{2}+\frac{2220544}{98543537118005}w^{5}t^{20}u^{3}+\frac{2678474752}{12317942139750625}w^{4}t^{24}+\frac{10944811008}{12317942139750625}w^{4}t^{23}u+\frac{983317504}{2463588427950125}w^{4}t^{22}u^{2}-\frac{239616}{98543537118005}w^{4}t^{21}u^{3}-\frac{66699264}{12317942139750625}w^{3}t^{25}-\frac{1160525824}{12317942139750625}w^{3}t^{24}u-\frac{376832}{11458550827675}w^{3}t^{23}u^{2}-\frac{120832}{492717685590025}w^{3}t^{22}u^{3}-\frac{2834432}{2463588427950125}w^{2}t^{26}+\frac{13172736}{1759706019964375}w^{2}t^{25}u+\frac{4431872}{2463588427950125}w^{2}t^{24}u^{2}-\frac{32768}{492717685590025}w^{2}t^{23}u^{3}+\frac{2162688}{12317942139750625}wt^{27}-\frac{4980736}{12317942139750625}wt^{26}u-\frac{65536}{2463588427950125}wt^{25}u^{2}-\frac{16384}{2463588427950125}t^{28}+\frac{131072}{12317942139750625}t^{27}u$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{24}{215}w^{7}+\frac{1009}{3010}w^{6}t+\frac{307}{3010}w^{6}u-\frac{1439}{3010}w^{5}t^{2}-\frac{389}{3010}w^{5}tu-\frac{2}{301}w^{5}u^{2}+\frac{831}{2107}w^{4}t^{3}+\frac{205}{2107}w^{4}t^{2}u+\frac{1}{86}w^{4}tu^{2}-\frac{15}{602}w^{4}u^{3}-\frac{402}{2107}w^{3}t^{4}-\frac{157}{2107}w^{3}t^{3}u-\frac{6}{2107}w^{3}t^{2}u^{2}+\frac{5}{301}w^{3}tu^{3}+\frac{622}{10535}w^{2}t^{5}+\frac{40}{2107}w^{2}t^{4}u+\frac{4}{301}w^{2}t^{3}u^{2}-\frac{10}{2107}w^{2}t^{2}u^{3}-\frac{108}{10535}wt^{6}-\frac{64}{10535}wt^{5}u-\frac{4}{2107}wt^{4}u^{2}+\frac{8}{10535}t^{7}+\frac{8}{10535}t^{6}u$ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.
