Equations satisfied by Igusa invariants of split Jacobians
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Polynomials
$n$ 
$\phi_n$
2:
768*i1^7 + 64*i1^6*i2 - 6912*i1^6 + 6336*i1^5*i2 + 786432*i1^5*i3 + 347328*i1^5 + 1152*i1^4*i2^2 + 25488*i1^4*i2 + 665321472*i1^4*i3 - 1492992*i1^4 + 48*i1^3*i2^3 + 12672*i1^3*i2^2 + 144506880*i1^3*i2*i3 + 1244160*i1^3*i2 - 9210101760*i1^3*i3 + 35831808*i1^3 + 3672*i1^2*i2^3 + 7372800*i1^2*i2^2*i3 + 362880*i1^2*i2^2 - 2600239104*i1^2*i2*i3 + 8957952*i1^2*i2 + 1962934272000*i1^2*i3^2 - 8408530944*i1^2*i3 + 324*i1*i2^4 + 31104*i1*i2^3 - 218972160*i1*i2^2*i3 + 746496*i1*i2^2 + 188743680000*i1*i2*i3^2 - 828112896*i1*i2*i3 - 271790899200*i1*i3^2 + 9*i2^5 + 864*i2^4 - 5529600*i2^3*i3 + 20736*i2^3 - 10616832*i2^2*i3 - 226492416000*i2*i3^2 + 1288490188800000*i3^3 - 5283615080448*i3^2
3:
-262144*i1^20+ 19660800*i1^19- 6029312*i1^18*i2- 759103488*i1^18- 589824*i1^17*i2^2+ 442368000*i1^17*i2+ 410365603086336*i1^17*i3+ 28251389952*i1^17- 9732096*i1^16*i2^2+ 71618713878528*i1^16*i2*i3- 13048086528*i1^16*i2- 39037497174392832*i1^16*i3- 643718430720*i1^16- 10911744*i1^15*i2^3+ 2701099008*i1^15*i2^2- 4486566525272064*i1^15*i2*i3+ 480385105920*i1^15*i2- 1019980674833533894656*i1^15*i3^2+ 1486944602890960896*i1^15*i3+ 15800429887488*i1^15- 552960*i1^14*i2^4+ 610467840*i1^14*i2^3+ 808394013278208*i1^14*i2^2*i3- 24425349120*i1^14*i2^2- 31541785025052672*i1^14*i2*i3- 7825129390080*i1^14*i2+ 76269293894454541811712*i1^14*i3^2- 40128585462580248576*i1^14*i3- 249333082951680*i1^14- 29859840*i1^13*i2^4+ 60461244481536*i1^13*i2^3*i3+ 2448506880*i1^13*i2^3- 101520233579151360*i1^13*i2^2*i3+ 2269452349440*i1^13*i2^2- 12871846084226511273984*i1^13*i2*i3^2+ 2271521383076855808*i1^13*i2*i3+ 192067825065984*i1^13*i2- 1732417880059570009669632*i1^13*i3^2+ 840664113370292551680*i1^13*i3+ 4420392857174016*i1^13- 7741440*i1^12*i2^5+ 4922726400*i1^12*i2^4- 7988703695732736*i1^12*i2^3*i3+ 595783434240*i1^12*i2^3- 1388854189686150660096*i1^12*i2^2*i3^2+ 3050129679961817088*i1^12*i2^2*i3- 1429003855872*i1^12*i2^2+ 1403517551078766293286912*i1^12*i2*i3^2- 5386471380332052480*i1^12*i2*i3- 1883305316597760*i1^12*i2+ 16673876967122993338461978624*i1^12*i3^3+ 17749563810661070492663808*i1^12*i3^2- 12521283529824515653632*i1^12*i3- 49322195910918144*i1^12- 276480*i1^11*i2^6+ 472780800*i1^11*i2^5+ 8818935201792*i1^11*i2^4*i3+ 39347804160*i1^11*i2^4+ 33146634862854144*i1^11*i2^3*i3+ 3758211717120*i1^11*i2^3+ 142464265537083722956800*i1^11*i2^2*i3^2- 49845990571521343488*i1^11*i2^2*i3+ 587246211366912*i1^11*i2^2+ 1595057872893078822822346752*i1^11*i2*i3^3- 56973825653331257444007936*i1^11*i2*i3^2+ 337514534352803856384*i1^11*i2*i3+ 34849971742703616*i1^11*i2- 2570981181974132655561243623424*i1^11*i3^3+ 15807802984090328604082176*i1^11*i3^2+ 161115160148411242512384*i1^11*i3+ 650384947198033920*i1^11- 10368000*i1^10*i2^6+ 7590100598784*i1^10*i2^5*i3+ 16230316032*i1^10*i2^5- 50611474686541824*i1^10*i2^4*i3+ 3685880920320*i1^10*i2^4- 4063037654207324749824*i1^10*i2^3*i3^2+ 5497277166523514880*i1^10*i2^3*i3+ 253564004302848*i1^10*i2^3- 6091786909372515688120320*i1^10*i2^2*i3^2+ 1388021238362277937152*i1^10*i2^2*i3+ 4219091766411264*i1^10*i2^2- 526013154332454008988981264384*i1^10*i2*i3^3+ 1573055753380653006606827520*i1^10*i2*i3^2+ 22080253398746186907648*i1^10*i2*i3- 163733039606661120*i1^10*i2- 2453388487998688008903673774080*i1^10*i3^4+ 133819375529919288361822558617600*i1^10*i3^3+ 9158582902078080770541355008*i1^10*i3^2- 1370841752076670218534912*i1^10*i3- 4841656529813766144*i1^10- 2626560*i1^9*i2^7+ 3479293440*i1^9*i2^6- 4927345624350720*i1^9*i2^5*i3+ 617647057920*i1^9*i2^5- 448816987892814446592*i1^9*i2^4*i3^2+ 2506125786388365312*i1^9*i2^4*i3+ 36268582809600*i1^9*i2^4+ 155714173231520179814400*i1^9*i2^3*i3^2+ 134317118356322254848*i1^9*i2^3*i3+ 1164162542469120*i1^9*i2^3- 38698587388405074434929459200*i1^9*i2^2*i3^3+ 278520971881648066501017600*i1^9*i2^2*i3^2- 9271657138820474732544*i1^9*i2^2*i3+ 58397671654686720*i1^9*i2^2+ 36099073282814741705843250561024*i1^9*i2*i3^3- 19926663371861089152046989312*i1^9*i2*i3^2- 260726695818064451076096*i1^9*i2*i3+ 2754005106457313280*i1^9*i2- 20231882274316146242800713266626560*i1^9*i3^4- 3051869989534184860798385003692032*i1^9*i3^3- 260969818049999175426043281408*i1^9*i3^2+ 9461652610002476001656832*i1^9*i3+ 47145486018234286080*i1^9- 77760*i1^8*i2^8+ 250076160*i1^8*i2^7- 217994731978752*i1^8*i2^6*i3+ 67287096576*i1^8*i2^6+ 251757491244761088*i1^8*i2^5*i3+ 10130711399424*i1^8*i2^5+ 4345037551856573743104*i1^8*i2^4*i3^2+ 5586285243524972544*i1^8*i2^4*i3+ 940502188523520*i1^8*i2^4- 1126549283812736306572689408*i1^8*i2^3*i3^3+ 26353065270756142441562112*i1^8*i2^3*i3^2+ 631300693429598552064*i1^8*i2^3*i3+ 47787744311377920*i1^8*i2^3+ 3639251417737919061672490696704*i1^8*i2^2*i3^3- 3619543697257394381028065280*i1^8*i2^2*i3^2+ 154306682903200574472192*i1^8*i2^2*i3+ 1090528245826191360*i1^8*i2^2- 2365840289313648427214306002599936*i1^8*i2*i3^4- 882354587111603906783043649536000*i1^8*i2*i3^3+ 153906790987378964481624244224*i1^8*i2*i3^2+ 4543779009838171966930944*i1^8*i2*i3+ 2634600689254268928*i1^8*i2+ 513130279329553908383820375725703168*i1^8*i3^4+ 39307295788162643528103646841536512*i1^8*i3^3+ 3515126149402773375029304360960*i1^8*i3^2- 31042057207982806511124480*i1^8*i3- 186363332966196707328*i1^8+ 2566080*i1^7*i2^8- 4707630710784*i1^7*i2^7*i3+ 9977478912*i1^7*i2^7+ 9959674815184896*i1^7*i2^6*i3+ 2383979765760*i1^7*i2^6- 2964191139303474069504*i1^7*i2^5*i3^2+ 3685126702505656320*i1^7*i2^5*i3+ 226382783053824*i1^7*i2^5+ 4261836152145340123840512*i1^7*i2^4*i3^2+ 687915020224265453568*i1^7*i2^4*i3+ 10352583298252800*i1^7*i2^4+ 129189157337235269677070942208*i1^7*i2^3*i3^3- 229194415569743967481233408*i1^7*i2^3*i3^2+ 33415633816536288854016*i1^7*i2^3*i3+ 228564234991042560*i1^7*i2^3- 63867289020732849807768630067200*i1^7*i2^2*i3^4- 82582594149563480389398206349312*i1^7*i2^2*i3^3+ 31809512209144747841965522944*i1^7*i2^2*i3^2- 302378671057261868089344*i1^7*i2^2*i3+ 2842595480511184896*i1^7*i2^2- 263445001358878845462050363959934976*i1^7*i2*i3^4+ 12642717775927462049417971372130304*i1^7*i2*i3^3- 637687038911057477038799585280*i1^7*i2*i3^2- 33937085042608806423429120*i1^7*i2*i3+ 66558333202213109760*i1^7*i2- 228754434455343352582726786015467405312*i1^7*i3^5+ 23020097304455386939021769266956337152*i1^7*i3^4- 290752597954881428801544723942604800*i1^7*i3^3- 27597149540143341543733938094080*i1^7*i3^2- 73467336673046785482031104*i1^7*i3+ 1277919997482491707392*i1^7- 388800*i1^6*i2^9+ 1183487760*i1^6*i2^8- 200617993175040*i1^6*i2^7*i3+ 300060039168*i1^6*i2^7- 127992397327226634240*i1^6*i2^6*i3^2+ 910704952329633792*i1^6*i2^6*i3+ 31683040026624*i1^6*i2^6+ 485361358449382181044224*i1^6*i2^5*i3^2+ 114426731939072311296*i1^6*i2^5*i3+ 1999879266631680*i1^6*i2^5- 2633904794803657666518319104*i1^6*i2^4*i3^3- 36631977183147739645476864*i1^6*i2^4*i3^2+ 1531882608471457136640*i1^6*i2^4*i3+ 92262135669719040*i1^6*i2^4+ 732722115302652378963932872704*i1^6*i2^3*i3^3+ 1103220154712546667820744704*i1^6*i2^3*i3^2- 218846577968433988632576*i1^6*i2^3*i3+ 3251523511469998080*i1^6*i2^3- 78312959509186210888479821568933888*i1^6*i2^2*i3^4+ 1431782753038613154970125761249280*i1^6*i2^2*i3^3- 167527591008567020878219444224*i1^6*i2^2*i3^2- 4989300743915910476070912*i1^6*i2^2*i3+ 73768819299119529984*i1^6*i2^2- 16382900952931562485090640564097908736*i1^6*i2*i3^5+ 28393397484790074063085192096845398016*i1^6*i2*i3^4- 104129714989680357214183627797037056*i1^6*i2*i3^3+ 1879430064611104442130873974784*i1^6*i2*i3^2+ 76366511174002065002201088*i1^6*i2*i3+ 745453331864786829312*i1^6*i2- 27060896728090295687680187327261391716352*i1^6*i3^5+ 25464986256455398227214337589003681792*i1^6*i3^4+ 1133173872274082174385990321920016384*i1^6*i3^3+ 158088696064240556956010648961024*i1^6*i3^2+ 726212968382690434165506048*i1^6*i3- 11664*i1^5*i2^10+ 73133280*i1^5*i2^9- 37271625596928*i1^5*i2^8*i3+ 25239589632*i1^5*i2^8+ 111537607680589824*i1^5*i2^7*i3+ 4191676563456*i1^5*i2^7+ 22590150707426302623744*i1^5*i2^6*i3^2+ 9300847147956043776*i1^5*i2^6*i3+ 422315122360320*i1^5*i2^6- 205247571556731690820829184*i1^5*i2^5*i3^3- 4771383285470347111956480*i1^5*i2^5*i3^2- 315542194361324273664*i1^5*i2^5*i3+ 26580009948807168*i1^5*i2^5+ 635994854576037958045245898752*i1^5*i2^4*i3^3+ 294148744565443512294703104*i1^5*i2^4*i3^2- 40778382976964414668800*i1^5*i2^4*i3+ 1010684566312058880*i1^5*i2^4- 6748562170189542313543732348059648*i1^5*i2^3*i3^4+ 23534747760916972418544672178176*i1^5*i2^3*i3^3+ 20813312356784609373601136640*i1^5*i2^3*i3^2- 665224565259006136811520*i1^5*i2^3*i3+ 21138433600332496896*i1^5*i2^3+ 7273400429232530368515762830444593152*i1^5*i2^2*i3^4- 14833641698925845401830658853044224*i1^5*i2^2*i3^3+ 1770712791298418603285620457472*i1^5*i2^2*i3^2+ 3703427929494353344462848*i1^5*i2^2*i3+ 186363332966196707328*i1^5*i2^2- 11155875296659977900762894463396900503552*i1^5*i2*i3^5- 234753746712516659364855809460959969280*i1^5*i2*i3^4+ 436957225259701418105091506193825792*i1^5*i2*i3^3+ 19438122462909192580734847549440*i1^5*i2*i3^2- 207188885209614717743529984*i1^5*i2*i3- 313448917305439382962893358684135665696768*i1^5*i3^6+ 1812418013578889828678548294357701825134592*i1^5*i3^5- 4976216186016423653320157988412905750528*i1^5*i3^4- 1656406006881902685050771824982360064*i1^5*i3^3- 499928481886698547970730362929152*i1^5*i3^2- 3203428509529231052488310784*i1^5*i3+ 1837080*i1^4*i2^10- 1060693180416*i1^4*i2^9*i3+ 2034434880*i1^4*i2^9+ 7769616101572608*i1^4*i2^8*i3+ 526055731200*i1^4*i2^8+ 222729812382628970496*i1^4*i2^7*i3^2+ 222601950561042432*i1^4*i2^7*i3+ 63068640215040*i1^4*i2^7- 63097858163161587253248*i1^4*i2^6*i3^2- 77491587448936857600*i1^4*i2^6*i3+ 4147379849134080*i1^4*i2^6+ 37894040689152442278938148864*i1^4*i2^5*i3^3+ 84434649797606963272482816*i1^4*i2^5*i3^2- 5417073913765036032000*i1^4*i2^5*i3+ 154972970916249600*i1^4*i2^5- 191435971508876714282352151363584*i1^4*i2^4*i3^4- 7648330658061138144111334785024*i1^4*i2^4*i3^3+ 5304865567158331526854213632*i1^4*i2^4*i3^2- 99425564359791520776192*i1^4*i2^4*i3+ 3100663091885506560*i1^4*i2^4+ 795515515089852018364221356494553088*i1^4*i2^3*i3^4- 714548277417818364782555399454720*i1^4*i2^3*i3^3+ 114586963921291074831982264320*i1^4*i2^3*i3^2- 172733501058891160485888*i1^4*i2^3*i3+ 25883796245305098240*i1^4*i2^3- 1218234228418314528560969762759044497408*i1^4*i2^2*i3^5- 79210756663801363176977931753624698880*i1^4*i2^2*i3^4+ 83382720406133725167118581718056960*i1^4*i2^2*i3^3- 5352252559188620473941696184320*i1^4*i2^2*i3^2+ 7213905716158942817550336*i1^4*i2^2*i3+ 767936980243981783192106758783275834015744*i1^4*i2*i3^5- 1118999235540370309403106293632859111424*i1^4*i2*i3^4- 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Definition
This table contains equations satisfied by the Igusa Invariants of genus $2$ curves with $(n,n)$ split Jacobians. More precisely, let $C$ be a genus $2$ curve, $Y^2 = f(X)$ a model thereof, $[I_2:I_4:I_6:I_8:I_{10}]$ be the Igusa invariants of this curve, and $(i_1,i_2,i_3)=(144 I_4/I_2^2,-1728(I_2 I_4-3 I_6)/I_2^3,486 I_{10}/I_2^5)$ the absolute invariants. If $\text{Jac}(C)$ is optimally $(n,n)$-split then its absolute invariants satisfy the equation $\phi_n=0$ with $\phi_n$ in this table.
Parameters
$n$
—   integer ($n \geq 2$)
Comments
(1)
More complete data is available at [4].
(2)
The case $n=2$ is classical, $n=3$ due to Kuhn [3], and $n=4$ due to Bruin and Doerksen [2].
References
[1]
Oskar Bolza, "Ueber die Reduction hyperelliptischer Integrale erster Ordnung und erster Gattung auf elliptische durch eine Transformation vierten Grades", Math. Ann. 28 (1887), no. 3, 447-456.
[2]
Nils Bruin, Kevin Doerksen, "The arithmetic of genus two curves with (4,4)-split Jacobians", Canad. J. Math. 63 (2011), 992-1021. (arXiv)
[3]
R. M. Kuhn, "Curves of genus 2 with split Jacobian". Trans. Amer. Math. Soc. 307(1988), no. 1, 41-49.
Links
Data properties
Entries are of type: integral polynomial
Table is complete: no
Sources of data: [4]