genus2isogenies

Sage code supporting

• [ arXiv ] Computing isogeny classes of typical principally polarized abelian surfaces over the rationals by Raymond van Bommel, Shiva Chidambaram, Edgar Costa, and Jean Kieffer

Computation of isogeny classes

The computation of isogeny classes from invariants depends be on

• SageMath
• pyhdme >= 0.0.6

  sage -pip install --upgrade git+https://github.com/edgarcosta/pyhdme

and optionally on:

• Pari/GP >= 2.15.0, precisely hyperellminimalmodel and hyperellred
• Magma >= 2.28-2, precisely HyperellipticCurveFromIgusaClebsch, ReducedWamelenModel to provide models with reasonable coefficient size.

Examples

See data/README.md for a description of the input/output.

A pair connected by an isogeny of degree 3^4

$ sage -python genus2isogenies.py "704:704.a:2159575534599616393:[[-1856,129536,-44,-2948]]:[[[-1,-1,-2,1,1,-2],[1,1,1,1]]]"
704:704.a:2159575534599616393:[[-1856,129536,-44,-2948]]:[[[-1,-1,-2,1,1,-2],[1,1,1,1]]]:[2,3]:[[-1856,129536,-44,-2948],[771904,687149056,260876,-197548868]]:[[[-1,-1,-2,1,1,-2],[1,1,1,1]],[[-2784,-2492,556,983,-50,-66,-6],[0,0,0,1]]]:[[0,9],[9,0]]

An example where we search for isogenies of degree 29^2 and 29^4

In this we recommend using higher number of threads, e.g., 12:

$ sage -python genus2isogenies.py "976:976.a:78262366554954104:[[1012,52318,-976,-37088]]:[[[1,2,0,-2,-1,-1],[0,0,1,1]]]" 12
976:976.a:78262366554954104:[[1012,52318,-976,-37088]]:[[[1,2,0,-2,-1,-1],[0,0,1,1]]]:[29]:[[1012,52318,-976,-37088]]:[[[1,2,0,-2,-1,-1],[0,0,1,1]]]:[[0]

An example of of square with isogenies of degree 3^4 and 5^4

$ sage -python genus2isogenies.py "277:277.a:639653774064676620:[[1408,49024,-277,-8864],[5482048,12835476224,277,-620480]]:[[[0,-1,-1],[1,1,1,1]],[[-6,11,-19,14,-9,1],[1]]]"
277:277.a:639653774064676620:[[1408,49024,-277,-8864],[5482048,12835476224,277,-620480]]:[[[0,-1,-1],[1,1,1,1]],[[-6,11,-19,14,-9,1],[1]]]:[3,5]:[[-4074752,3079190656,-53383717,155880885760],[1408,49024,-277,-8864],[5482048,12835476224,277,-620480],[5786446528,440253117070336,-3537013,285292529056]]:[[[-10406,-10383,5266,7578,169,-1396,-252],[0,0,0,1]],[[0,-1,-1],[1,1,1,1]],[[-6,11,-19,14,-9,1],[1]],[[-14580,51520,-63684,31054,-3823,-637,-18],[1]]]:[[0,25,0,9],[25,0,9,0],[0,9,0,25],[9,0,25,0]]

and now without reducing the models:

$ sage -python genus2isogenies.py "277:277.a:639653774064676620:[[1408,49024,-277,-8864],[5482048,12835476224,277,-620480]]:[[[0,-1,-1],[1,1,1,1]],[[-6,11,-19,14,-9,1],[1]]]" 1 0 0
277:277.a:639653774064676620:[[1408,49024,-277,-8864],[5482048,12835476224,277,-620480]]:[[[0,-1,-1],[1,1,1,1]],[[-6,11,-19,14,-9,1],[1]]]:[3,5]:[[-4074752,3079190656,-53383717,155880885760],[1408,49024,-277,-8864],[5482048,12835476224,277,-620480],[5786446528,440253117070336,-3537013,285292529056]]:[[[-358788700398291516766687458235974560914616733718839868617824,-357995682897891679664473945691343827212806606400414602715632,181566528569806181750276393914149724944875237340323923519264,261282026870868067851043394052682608361614992131594130730912,5826954676850976968438418262721478069822239861472605976976,-48132714372094460638698413578456706422910336370507443454784,-8680093727202564803576164483077704757856502278850464820732],[]],[[0,-1,-1],[1,1,1,1]],[[-6,11,-19,14,-9,1],[1]],[[-17588459929395102649944419294343078237087543039801061629236836683347243600,20427513364465449728818904641828577313131489378882607185354614789636913000,-8393322150583180805390938327715976335874632430018392406844028984632688200,1353205011709289609225327763793294838113490509389129389918960714769082000,-41850011666885775142989074775546619177508931977339604845350537952215200,-5308650305003134662393957918656152954849370614183853269112133077924600,-81951719974319403021519075931227061052563182723249878939981471185800],[]]]:[[0,25,0,9],[25,0,9,0],[0,9,0,25],[9,0,25,0]]

An example with isogeny graph of size 42

sage -python genus2isogenies.py "2084783416934400:None:0:[[12279421902241,27720894398734429039,-425954713866501456000000,3620409974473702743642699000000]]:[[[2288,-14840,12679,37445,11208,720],[]]]"
2084783416934400:None:0:[[12279421902241,27720894398734429039,-425954713866501456000000,3620409974473702743642699000000]]:[[[2288,-14840,12679,37445,11208,720],[]]]:[2]:[[-3748458025135679,694225931974496339453119,619920649486901346282562500,30351145841140999930007171039437500],[-575638324559759,12286578492875215682839,13102016470383120462000,-224839864777508695166428548000],[-406141282135679,3744876437299379753119,29722918120155643304391992437500,-1217542684354332972753303259813664437500],[-80383490035679,12397649316423490763119,-44463027549129080104213745062500,544326914049434484048879587803148062500],[-39130396949519,224771934218746336471,33504097249407565334531250000,-22901535216381088086967495237500000],[-29736646361519,237523650827887409671,-12179105721334563475668750000,363917869213050132126420492562500000],[-24969123696959,2414177579327779329281,-541688943980706749807580000,-16675118661876865484907715025625000],[-24387851335679,148304114802468233119,42367380329452423095703125000,-95051345113366815740778808593750000],[-20449399330559,51062407712928219041,-7213951625251464843750,-45971304847383611623535156250],[-15791884530719,215590247115509882609,47356053770269104960937500000,267209078469608433644877041015625000],[-4297028635679,273245445614272103119,-31086555719505380859375000000,336139610899237512350800781250000000],[-2643503216399,2268775635162902551,11546728087630519379232000,-85635963507759515483489729262000],[-2050725512399,2503393124571084151,-32808309228897901590528000,10527906496380473534819292288000],[-2009059703519,928052307259830525871,4370829531085326660316875000,-132043020094452069528753363266250000],[-172636530239,36481379719206079,10990213851046510080000,-20351116098435454214056200000],[90869596801,2892831627474105569,14353334880907751993013750,109870671855133255206080231606250],[777877744561,414540923458135159,-366533137580433408000,780690217311844677631872000],[1816339826401,8432560063739880271,19590675599909872252080000,-19187138391597119845216063125000],[5021358420241,11355170114352267161,-18180573015720211200000000,43898147778895338596382000000000],[5109519582241,10636589947593682961,9739924157430697230000000,15877515421679104665502363125000],[5953059888241,6681824907861688361,3579464269632150900000000,29152227758902247320950768750000],[6238363166401,7938132872141753729,3236090384489120196480000,26451732945969965354172683400000],[9611055255601,46287891979495111351,391702997252051856949500,-3204702406547240661097032874500],[12279421902241,27720894398734429039,-425954713866501456000000,3620409974473702743642699000000],[39692690076481,331597027847041232129,-35818527818850527123919000000,231576669217926136204209911646000000],[65296985676481,79686679571018607871,-14234483439565966811359125000,215548932399207542218235190489750000],[66229041888241,966123794833267480361,-3770751561810925268769192000,-121242671841971574307516832737398000],[78382940363041,756916712213286003281,-4269947122294874733733020000,192145443498572255960645822757975000],[80953658652241,681355493422927507961,19342799175023763549311448000,-144964503820539017136574047507798000],[156378318576481,2891757867604992417871,668960409796861607461968750,-548321236899155371003203061031250],[171444930234481,2476832707269627554071,2374777240299077243175000000,-24539275058581066566182900906250000],[176347394664241,2341367101973956253239,-720057292073437500,10779012088755395976562500],[181261323514321,2204992895776892018119,-2142006574417084019531250000,38820796857202495975770023437500000],[318773367269761,32381926358083761671009,-15752110470557433151245117187500,-868638502514177650580196769409179687500],[716020592969761,9259817782446032741009,39068667020133440886840820312500,-479963258886280000703213600463867187500],[749182931689441,20205776948886980402959,-2628609975788326313355000,107301661049562895558152391050000],[1280229737149441,45784496937026387091041,133448568384332221020000,10959940733681371231185600825000],[1290253363788241,45030776685898747426361,2281294577820526632000,187475809071605186449616598000],[2090861907556801,229217014882869168324799,259860698678255081176757812500,-11734750191947142539262771606445312500],[2821415499364321,149863425020255723903119,-12854254237976074218750,687930909813884731750488281250],[3566441356769761,64594205130502291878991,-208386780734661849975585937500,20659430027572340314516957397460937500],[10787852211364321,967263215211677497896881,62734725963892505710687500,57993315830416929896354365476187500]]:[[[65547662,-186088592,128433191,64326466,-83931727,-1638800,13351800],[]],[[20593634,56779779,20285401,-60561897,-43318165,15834654,14486913],[1,0,1]],[[-5992051697036548,1766062200083948,5201837126089951,-100633285331414,-918782055369447,-207793540033740,85013055928350],[]],[[5556759752250,-788738413680,4612486177957,724970578378,-33025768197,13086749880],[]],[[-87056860870,3419187121,19107730,72082,43787,160],[0,1]],[[87657000,74954615,10981348,-8268298,-1419613,540920,-29200],[0,1]],[[94526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Heuristically verifying the computation

It depends on CHIMP and smalljac (no assert failing means success)

magma -b input:="704:704.a:2159575534599616393:[[-1856,129536,-44,-2948]]:[[[-1,-1,-2,1,1,-2],[1,1,1,1]]]:[2,3]:[[-1856,129536,-44,-2948],[771904,687149056,260876,-197548868]]:[[[0,4,4,-1,-2],[0,0,0,1]],[[-1140,-5166,-4646,2135,-236,4],[1]]]:[[0,9],[9,0]]" verify.m