Elliptic curves with CM defined
over extensions of type (2,...,2)

Joan-C. Lario

This "horizontal" page contains data on the finite set (up to isomorphism) of elliptic curves with complex multiplication defined over extensions of type (2,...,2). An elliptic curve over the complex numbers is said to have CM if its algebra of endomorphisms is larger than the ring of rational integers. Class Field theory provides a framework to the study of CM elliptic curves, so that to classify the j-invariants of those defined over extensions of type (2,...,2) all what is needed is the complete list of quadratic discriminants (fundamental or non-fundamental) of one class per genus. In other words, we provide the complete list of j-invariants with CM such that the extension Q(j)/Q is Galois, which is equivalent to say that Q(j) is totally real.

The homely computations use Pari and were motivated to discard CM cases among the elliptic Q-curves defined over extensions of type (2,...,2).

Notations:

K is an imaginary quadratic field, and O_K its ring of integers;
E is a CM elliptic curve by the order O_f = Z+ f × O_K, and j = j(E) its j-invariant;
h = #Cl(O_f) is the class number of the order.

The elements of the lists h_2^i are of the form {D,{d_1,...,d_i},P(x)}, where:

K = Q(sqrt(D));
Q(j) = Q(sqrt(d_1),...,sqrt(d_i));
P(x) = Irr(j,Q;x).

Idem for the elements of the lists h_2^i_order but D = disc(K) × f^2 being non-fundamental.

h_2=
{ -15,{5},x^2+191025*x-121287375},
{ -20,{5},x^2-1264000*x-681472000},
{ -24,{2},x^2-4834944*x+14670139392},
{ -35,{5},x^2+117964800*x-134217728000},
{ -40,{5},x^2-425692800*x+9103145472000},
{ -51,{17},x^2+5541101568*x+6262062317568},
{ -52,{13},x^2-6896880000*x-567663552000000},
{ -88,{2},x^2-6294842640000*x+15798135578688000000},
{ -91,{13},x^2+10359073013760*x-3845689020776448},
{-115,{5},x^2+427864611225600*x+130231327260672000},
{-123,{41},x^2+1354146840576000*x+148809594175488000000},
{-148,{37},x^2-39660183801072000*x-7898242515936467904000000},
{-187,{17},x^2+4545336381788160000*x-3845689020776448000000},
{-232,{29},x^2-604729957849891344000*x+14871070713157137145512000000000},
{-235,{5},x^2+823177419449425920000*x+11946621170462723407872000},
{-267,{89},x^2+19683091854079488000000*x+531429662672621376897024000000},
{-403,{13},x^2+2452811389229331391979520000*x-108844203402491055833088000000},
{-427,{61},x^2+15611455512523783919812608000*x+155041756222618916546936832000000}

h_2_order=
{-32,{2},x^2-52250000*x+12167000000},
{-36,{3},x^2-153542016*x-1790957481984},
{-48,{3},x^2-2835810000*x+6549518250000},
{-60,{5},x^2-37018076625*x+153173312762625},
{-64,{2},x^2-82226316240*x-7367066619912},
{-72,{6},x^2-377674768000*x+232381513792000000},
{-75,{5},x^2+654403829760*x+5209253090426880},
{-99,{33},x^2+37616060956672*x-56171326053810176},
{-100,{5},x^2-44031499226496*x-292143758886942437376},
{-112,{7},x^2-274917323970000*x+1337635747140890625},
{-147,{21},x^2+34848505552896000*x+11356800389480448000000}

h_4=
{-84,{3,7},x^4-3196800946944*x^3-5663679223085309952*x^2+88821246589810089394176*x-5133201653210986057826304},
{-120,{2,5},x^4-883067971104000*x^3+26329406807264910336000*x^2-2588458316335175909376000000*x+4934510722321469030006784000000},
{-132,{3,11},x^4-4736863498464000*x^3-325211610485778048000000*x^2+54984539729717250048000000000*x+1656636925108948992000000000000},
{-168,{6,14},x^4-483435712076832000*x^3+336511679671210230144000000*x^2-264691184105480095991808000000000*x+496644064976895846912000000000000000},
{-195,{5,13},x^4+11284411506057216000*x^3+25349140792043819237376000*x^2+104773100319600336175104000000*x-233490285492432753672585216000000},
{-228,{3,19},x^4-399605224650084576000*x^3-7985216535621460489954944000000*x^2+58827548670433207062445836288000000000*x+120020259495560805847424176128000000000000},
{-280,{2,5},x^4-67667966893419063840000*x^3+17602516524144666384420962098176000*x^2-708555761206745670461365038563328000000*x+1775168961518724506399346503073398784000000},
{-312,{2,13},x^4-1258031100283439093280000*x^3+1411168483733488619338991640960000000*x^2-152340504750882110373595179663329280000000000*x+1698899690981885675579246225669492736000000000000},
{-340,{5,17},x^4-14383245771217510630675200*x^3-54548817402421378465247510316573696000*x^2+5906485031594874833231597894020684185600000*x+43039377624755967291385639037347037184000000},
{-372,{3,31},x^4-206603714804587147622880000*x^3-2969541010382978868435960918595200000000*x^2+1755509254864401819594526832548625909760000000000*x+41393149892607462736698558825033501904896000000000000},
{-408,{2,17},x^4-3622859125108878497350176000*x^3+218066148024051247931306674050097536000000*x^2-334918514756463762318006309600841904719872000000000*x+13375974716483932888129605820405217248677888000000000000},
{-435,{5,29},x^4+28597298728131202056826060800*x^3+87465379468169320817492479772196864000*x^2+42866222697779107335351550466659555737600000*x-12512019875237835915942574589201734434816000000},
{-483,{21,69},x^4+966618711103413979025620992000*x^3+9557426544972522152310585774047232000000*x^2+160587932046974848398336021151875072000000000*x-296241507936739247491345278560108544000000000000},
{-520,{5,13},x^4-12958889442406058296422344736000*x^3+46650003139146307922421888174845453223975936000*x^2-78006534528871949845908360976579586206001479680000000*x+171517475891022372428505519185548559222346497654784000000},
{-532,{7,19},x^4-29478909019098139074177479136000*x^3-160054212938390343773833947283393690785408000000*x^2+5131537740610192962070880163006969643272192000000000*x-19077542993352945680961028994697271308288000000000000},
{-555,{5,37},x^4+138859536630220704987259502592000*x^3+7191013406366483381037450688276469907456000*x^2+19282254568556435196991625190065063388512256000000*x-532755731205331063356397364951543957176713216000000},
{-595,{5,17},x^4+1908606683491595666107623383040000*x^3+8752111455147508300981595950899265536000*x^2+483054636550112292687021684688517332992000000*x-91399742601830803813322386656934773129216000000},
{-627,{33,57},x^4+14586137722924213400310156521472000*x^3+3563858169242172480409901737583233204224000000*x^2+526326624169690832922357632213666758656000000000*x-1261687189208313891495979730091871567872000000000000},
{-708,{3,59},x^4-2012303924332635494819557244440800000*x^3-2854565250565963840094617979015298078098347812480000000*x^2+3603887011528002652771717224491220641587422892784070051840000000000*x+4046686423378034814414234559373865948538701215210194862739456000000000000},
{-715,{5,13},x^4+3038922093329613647424771157499904000*x^3+60156378344564221943954774472086041657344000*x^2+94657547256854352451418607502680693669888000000*x+13189879204176058896562640516998642620432384000000},
{-760,{2,5},x^4-41045008988631123111685822548134227200*x^3+262960509575258849119050573504013616920976671774792704000*x^2-8762694788548498478760416933120597566268079681131589510758400000*x+57390991709103678336339431944416743303984993656228540622045184000000},
{-795,{5,53},x^4+294853904675299611949375562546552832000*x^3+1962512368737475150054890329369747830206508302336000*x^2+96989374802114211792220362019627433906928110027145216000000*x+1580866394929445594613317271657673734190830966521462784000000},
{-1012,{11,23},x^4-25320300665394312513202440887044222231008000*x^3-127409933077387882483393397275452601210672521773684542789248000000*x^2+1243508019466325039942928040075544459231301705571821435488256000000000*x-204344290478354698106731378125784194576718103432833630334976000000000000},
{-1435,{5,41},x^4+4835907878329132222450395857259654466718969836339200*x^3+105161502065491843193116493512870644772977750388041962225664000*x^2+2539970793779946148723473087883821948343643667859030533390886502400000*x-4009811510734177961140258455491639109235283276327883374802990268416000000}
h_4_order=
{-96,{2,3},x^4-23340144296736*x^3+670421055192156288*x^2+447805364111967209472*x-984163224549635621646336},
{-160,{2,5},x^4-181195519824640800*x^3-2940735389875294896000*x^2+13208221536779701382400000*x-293835053960432980416000000},
{-180,{3,5},x^4-2018504138609120000*x^3-2867757758882006477169664000*x^2+16660473763558887652278272000000*x-6776421923145961044033929216000000},
{-192,{2,3},x^4-8041801037378436000*x^3+15705521635909735050750000*x^2+826335556188178615474500000000*x-1080060886113159937649308593750000},
{-240,{3,5},x^4-1370337635584848362400*x^3+15510636623637225985530243375*x^2-3213137488352330508627918491550000*x+51848746810441819437662737568196890625},
{-288,{2,3},x^4-142637765058468510772000*x^3-87330008255955399131086000000*x^2+136478143044657426076564000000000*x+40994594700208456153393000000000000},
{-315,{5,21},x^4+1641640623633325773619200*x^3+116880346596078822090604544000*x^2+256755031834901461034493542400000*x+16319518649540946810407747584000000},
{-352,{2,11},x^4-39625012256717991533364000*x^3-99328134588221245548396654000000*x^2+149685745558232337507189972000000000*x+233814863662235465757810993000000000000},
{-448,{2,7},x^4-75579535015741588088518020000*x^3-1251995474985759392628697477841250000*x^2+18314847446238545696830716579562500000000*x-8964424282273362890505339044524081787109375},
{-928,{2,29},x^4-365698321891389219219142531076614125108000*x^3-8992980876865725569995845414882451608653638542000000*x^2+13390564998216290984073146179239751905642859284000000000*x+221147287295840631009155378215919896241234993293617000000000000}
h_8=
{-420,{3,5,7},x^8-9149442836345200514625984000*x^7-875149953790588830141513408467102803968000*x^6+1830759213424837201179800381566577321252585472000000*x^5-654322371647418124227660658480955270871337023307776000000*x^4-25522767602086427249222314481632218756271917729054720000000000*x^3+343101579425550237027264306100046973285019210099445465088000000000*x^2+3322057458803342041196516880396913435501050824830746624000000000000*x-6774993267880302065499271589077244607702692106378720313344000000000000},
{-660,{3,5,11},x^8-112580501740220781071235722688652800*x^7-37774101659997423005408109970789812947961704097792000*x^6+18239982511248258426444458529164454527146102008187866100531200000*x^5-174143430251936272102859349073539136097610895959350576901027004416000000*x^4-129459690207405843746931738609689674381742368424208527099314604448153600000000*x^3+145752537220577621057671362142160834897197172505227599866968670493212672000000000*x^2+802614337824902100615094698396471270494041411707339702383464010676279705600000000000*x-956825222980306326585980265538528042697197583016225087193752883841407647744000000000000},
{-840,{5,6,21},x^8-3494487845306481075093315600749304691200*x^7+206573882876758009898241769258678546966352946154161788928000*x^6-3134769336133353615460866275393209275783941494973163498240275428147200000*x^5+267678830160178923896641219852982233572924885080172883621331723095220158464000000*x^4-1111712812272489788109971969097031933551408742194642794550538731744862298072678400000000*x^3+454668527671405657965710869144455214652592634921420559367890411545189775674863255552000000000*x^2-5112159939990146378938499680802637042771646067107417706535388782137560566356569069977600000000000*x+7587169380271379738636919142674280077130439504327732605512510089785122099137867107270656000000000000},
{-1092,{3,7,13},x^8-1220096358687753260204078826101522741473728000*x^7-42617815331984530664240019051172769446368789029954440306166016000000*x^6+45539601715292538862264980052027187457221547030758023509730728512783118336000000000*x^5+1362719999058592388335992752851672703626579848779924470931866989317688863514624000000000000*x^4-4114380912031888144184139782379675690399375140970078847729371868187444847401238528000000000000000*x^3+14147331105803618441787960284346172301266050546136888382416860722817545313985757184000000000000000000*x^2+7812160693352835962330966127506219038319484257784013790202419396792420884688142336000000000000000000000*x-11399327725494218798737808488898891371555857163161203882074016958512319065701220352000000000000000000000000},
{-1155,{5,21,33},x^8+23373692778731029019776599932268751032999936000*x^7+66829784571649628051957299392021041534581719969792169869312000*x^6+125802782591288988289501749810981813757267278571203492224824246272000000*x^5+529969159795786169479551056606836273591438303269788897135324287419285504000000*x^4+7711851888998550611415837895667386487366740340143356218759041649486069760000000000*x^3-9518746549833191407809353203709672284413276783319410082239757655160643387392000000000*x^2-8900615215725314134559469276744109057080931256308383138559489050825505898496000000000000*x+83347851037168500148480644982979700827573455605987270779845921763214445510656000000000000},
{-1320,{5,6,22},x^8-37175310962524209402711325090684825397858189376000*x^7+226663670534475149680772726035193159902547895822786263994560472042657792000*x^6-7564848369936329757748832158280770921137664340570303183861640564358996428548474241024000000*x^5+63448588042551212367314599642690745398746089936036428003699736173360732007940703292533161590784000000*x^4-11346725453301019241313454803369930713744430991711423923070736653179704868794839532252190334016028672000000000*x^3+1440564325039592709845067874877463044088141586574151125929003856830996185412289341320527503606330769276928000000000*x^2-38039887135901201776740554544943720383598953324261921806673528172358159163524934578063481550048308706148352000000000000*x+97393688378575070484102609772364748569437702720422138019614918281329114615938477076822153891382147228847046656000000000000},
{-1380,{3,5,23},x^8-483428242454398634463543323663456033879150969280000*x^7-10629137536424828578817878065952562448875253668798815309065879486385563648000*x^6+834210097027380610396492239537077265364446379760873874291185809535255620499245325451264000000*x^5-11198719491880226077458851730219954432641563194866892969956474394228580008323634692847376865427456000000*x^4-3204053117604554451212581049183434081694096098503753773217096642404791036731056991709579306948808933376000000000*x^3-359450962870172070183849496024185547533350025952232695319442086650832640411793554090144649654068466818220032000000000*x^2+1515672488750192406755322903951442203568732081884436967876801416160928484351453665061090181618261938610896896000000000000*x-1043717434674580402717419972320183214179349425352220925666143898343059593202965992203361417535941416422257721344000000000000},
{-1428,{3,7,17},x^8-3616297983752590489170592931739244329043525804992000*x^7-217468476883849476852145600742611039750726891957499097582953392796739840000000*x^6+33379850139585632155557058810145064423499658538072264572138227867514574381239272501248000000000*x^5+12303208205702120460701668812255683079036859290282157017127878896492104050452329313939251200000000000000*x^4-303386981784522159659729731008790456216922186724680934591752863711575979381177651000980311113728000000000000000*x^3-187852796407105234735480351477265459845155497818612689401274425061640747631666840894423310532608000000000000000000*x^2+3456839501734300594235950152209592698750414128946811077602647076913638860503714431187067863040000000000000000000000000*x-3711489744391505840909265987948679379016518570689838859089537543927618911465409927842627584000000000000000000000000000000},
{-1540,{5,7,11},x^8-348332834308833141926080575603441539337630931347494400*x^7-205585092816139405052854352895728829746651659435621725196337548174692722092032000*x^6+10513992728337323857474607080214044088440240805433393995398011461565738649957200239001600000*x^5-466360079075241326995156571883638104758296725599974536329327058285603153961495639931946008576000000*x^4-73560169330053609039053599079825580643757504375974892764340262860363008316091765380340603735244800000000*x^3+7427862043816793225887839351921172697648972542206102972589825511255238923000003268762243855770714112000000000*x^2+46787441786732458820488281906991541116619282261319097960218235215598040810357936414042690370937867468800000000000*x+35393772430126097929846875535887051112083338397643764824010316117729858709536249792898603024023289856000000000000},
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