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Properties

Label 44.44.1588950548...0625.1
Degree $44$
Signature $[44, 0]$
Discriminant $5^{22}\cdot 89^{43}$
Root discriminant $179.71$
Ramified primes $5, 89$
Class number Not computed
Class group Not computed
Galois group $C_{44}$ (as 44T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12360161581, 146388138349, -452559496938, -2357998728050, 5275330691163, 15665394240053, -30095367088880, -57361851429233, 99728212374110, 130979137701417, -211144037309930, -200381975387743, 303617014545478, 215575931040434, -309586495847682, -168838600437028, 231114721012025, 98733171381559, -129393380339543, -43911599382957, 55312614944324, 15048715494929, -18288247142006, -4008528403014, 4717270351997, 833885620680, -953714610433, -135618356054, 151268636295, 17194056330, -18764844858, -1686827784, 1806620424, 126368145, -133171784, -7074358, 7352544, 285780, -293502, -7851, 7981, 131, -132, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 132*x^42 + 131*x^41 + 7981*x^40 - 7851*x^39 - 293502*x^38 + 285780*x^37 + 7352544*x^36 - 7074358*x^35 - 133171784*x^34 + 126368145*x^33 + 1806620424*x^32 - 1686827784*x^31 - 18764844858*x^30 + 17194056330*x^29 + 151268636295*x^28 - 135618356054*x^27 - 953714610433*x^26 + 833885620680*x^25 + 4717270351997*x^24 - 4008528403014*x^23 - 18288247142006*x^22 + 15048715494929*x^21 + 55312614944324*x^20 - 43911599382957*x^19 - 129393380339543*x^18 + 98733171381559*x^17 + 231114721012025*x^16 - 168838600437028*x^15 - 309586495847682*x^14 + 215575931040434*x^13 + 303617014545478*x^12 - 200381975387743*x^11 - 211144037309930*x^10 + 130979137701417*x^9 + 99728212374110*x^8 - 57361851429233*x^7 - 30095367088880*x^6 + 15665394240053*x^5 + 5275330691163*x^4 - 2357998728050*x^3 - 452559496938*x^2 + 146388138349*x + 12360161581)
 
gp: K = bnfinit(x^44 - x^43 - 132*x^42 + 131*x^41 + 7981*x^40 - 7851*x^39 - 293502*x^38 + 285780*x^37 + 7352544*x^36 - 7074358*x^35 - 133171784*x^34 + 126368145*x^33 + 1806620424*x^32 - 1686827784*x^31 - 18764844858*x^30 + 17194056330*x^29 + 151268636295*x^28 - 135618356054*x^27 - 953714610433*x^26 + 833885620680*x^25 + 4717270351997*x^24 - 4008528403014*x^23 - 18288247142006*x^22 + 15048715494929*x^21 + 55312614944324*x^20 - 43911599382957*x^19 - 129393380339543*x^18 + 98733171381559*x^17 + 231114721012025*x^16 - 168838600437028*x^15 - 309586495847682*x^14 + 215575931040434*x^13 + 303617014545478*x^12 - 200381975387743*x^11 - 211144037309930*x^10 + 130979137701417*x^9 + 99728212374110*x^8 - 57361851429233*x^7 - 30095367088880*x^6 + 15665394240053*x^5 + 5275330691163*x^4 - 2357998728050*x^3 - 452559496938*x^2 + 146388138349*x + 12360161581, 1)
 

Normalized defining polynomial

\( x^{44} - x^{43} - 132 x^{42} + 131 x^{41} + 7981 x^{40} - 7851 x^{39} - 293502 x^{38} + 285780 x^{37} + 7352544 x^{36} - 7074358 x^{35} - 133171784 x^{34} + 126368145 x^{33} + 1806620424 x^{32} - 1686827784 x^{31} - 18764844858 x^{30} + 17194056330 x^{29} + 151268636295 x^{28} - 135618356054 x^{27} - 953714610433 x^{26} + 833885620680 x^{25} + 4717270351997 x^{24} - 4008528403014 x^{23} - 18288247142006 x^{22} + 15048715494929 x^{21} + 55312614944324 x^{20} - 43911599382957 x^{19} - 129393380339543 x^{18} + 98733171381559 x^{17} + 231114721012025 x^{16} - 168838600437028 x^{15} - 309586495847682 x^{14} + 215575931040434 x^{13} + 303617014545478 x^{12} - 200381975387743 x^{11} - 211144037309930 x^{10} + 130979137701417 x^{9} + 99728212374110 x^{8} - 57361851429233 x^{7} - 30095367088880 x^{6} + 15665394240053 x^{5} + 5275330691163 x^{4} - 2357998728050 x^{3} - 452559496938 x^{2} + 146388138349 x + 12360161581 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $44$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[44, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1588950548018177733817067321623614819860239312933539604872830957860279621432150312826633453369140625=5^{22}\cdot 89^{43}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $179.71$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(445=5\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{445}(256,·)$, $\chi_{445}(1,·)$, $\chi_{445}(129,·)$, $\chi_{445}(9,·)$, $\chi_{445}(266,·)$, $\chi_{445}(11,·)$, $\chi_{445}(271,·)$, $\chi_{445}(16,·)$, $\chi_{445}(401,·)$, $\chi_{445}(146,·)$, $\chi_{445}(406,·)$, $\chi_{445}(409,·)$, $\chi_{445}(284,·)$, $\chi_{445}(34,·)$, $\chi_{445}(424,·)$, $\chi_{445}(156,·)$, $\chi_{445}(176,·)$, $\chi_{445}(49,·)$, $\chi_{445}(306,·)$, $\chi_{445}(309,·)$, $\chi_{445}(311,·)$, $\chi_{445}(441,·)$, $\chi_{445}(314,·)$, $\chi_{445}(69,·)$, $\chi_{445}(199,·)$, $\chi_{445}(331,·)$, $\chi_{445}(79,·)$, $\chi_{445}(81,·)$, $\chi_{445}(339,·)$, $\chi_{445}(84,·)$, $\chi_{445}(214,·)$, $\chi_{445}(121,·)$, $\chi_{445}(91,·)$, $\chi_{445}(186,·)$, $\chi_{445}(94,·)$, $\chi_{445}(144,·)$, $\chi_{445}(99,·)$, $\chi_{445}(109,·)$, $\chi_{445}(111,·)$, $\chi_{445}(374,·)$, $\chi_{445}(169,·)$, $\chi_{445}(249,·)$, $\chi_{445}(251,·)$, $\chi_{445}(381,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $\frac{1}{179} a^{38} + \frac{26}{179} a^{37} - \frac{8}{179} a^{36} - \frac{39}{179} a^{35} + \frac{46}{179} a^{34} - \frac{36}{179} a^{33} - \frac{49}{179} a^{32} + \frac{39}{179} a^{31} - \frac{44}{179} a^{30} - \frac{69}{179} a^{29} - \frac{24}{179} a^{28} - \frac{59}{179} a^{27} + \frac{82}{179} a^{26} + \frac{25}{179} a^{25} + \frac{12}{179} a^{24} + \frac{66}{179} a^{23} + \frac{81}{179} a^{22} - \frac{77}{179} a^{21} + \frac{79}{179} a^{20} - \frac{5}{179} a^{19} - \frac{76}{179} a^{18} + \frac{18}{179} a^{17} - \frac{34}{179} a^{16} + \frac{22}{179} a^{15} + \frac{26}{179} a^{14} - \frac{49}{179} a^{13} - \frac{44}{179} a^{12} - \frac{68}{179} a^{11} + \frac{62}{179} a^{10} - \frac{11}{179} a^{9} + \frac{82}{179} a^{8} - \frac{67}{179} a^{7} + \frac{1}{179} a^{6} - \frac{61}{179} a^{5} - \frac{16}{179} a^{4} - \frac{11}{179} a^{3} + \frac{42}{179} a^{2} - \frac{86}{179} a - \frac{40}{179}$, $\frac{1}{179} a^{39} + \frac{32}{179} a^{37} - \frac{10}{179} a^{36} - \frac{14}{179} a^{35} + \frac{21}{179} a^{34} - \frac{8}{179} a^{33} + \frac{60}{179} a^{32} + \frac{16}{179} a^{31} + \frac{1}{179} a^{30} - \frac{20}{179} a^{29} + \frac{28}{179} a^{28} + \frac{5}{179} a^{27} + \frac{41}{179} a^{26} + \frac{78}{179} a^{25} - \frac{67}{179} a^{24} - \frac{24}{179} a^{23} - \frac{35}{179} a^{22} - \frac{67}{179} a^{21} + \frac{89}{179} a^{20} + \frac{54}{179} a^{19} + \frac{25}{179} a^{18} + \frac{35}{179} a^{17} + \frac{11}{179} a^{16} - \frac{9}{179} a^{15} - \frac{9}{179} a^{14} - \frac{23}{179} a^{13} + \frac{2}{179} a^{12} + \frac{40}{179} a^{11} - \frac{12}{179} a^{10} + \frac{10}{179} a^{9} - \frac{51}{179} a^{8} - \frac{47}{179} a^{7} - \frac{87}{179} a^{6} - \frac{41}{179} a^{5} + \frac{47}{179} a^{4} - \frac{30}{179} a^{3} + \frac{75}{179} a^{2} + \frac{48}{179} a - \frac{34}{179}$, $\frac{1}{179} a^{40} + \frac{53}{179} a^{37} + \frac{63}{179} a^{36} + \frac{16}{179} a^{35} - \frac{48}{179} a^{34} - \frac{41}{179} a^{33} - \frac{27}{179} a^{32} + \frac{6}{179} a^{31} - \frac{44}{179} a^{30} + \frac{88}{179} a^{29} + \frac{57}{179} a^{28} - \frac{40}{179} a^{27} - \frac{40}{179} a^{26} + \frac{28}{179} a^{25} - \frac{50}{179} a^{24} + \frac{1}{179} a^{23} + \frac{26}{179} a^{22} + \frac{47}{179} a^{21} + \frac{32}{179} a^{20} + \frac{6}{179} a^{19} - \frac{39}{179} a^{18} - \frac{28}{179} a^{17} + \frac{5}{179} a^{16} + \frac{3}{179} a^{15} + \frac{40}{179} a^{14} - \frac{41}{179} a^{13} + \frac{16}{179} a^{12} + \frac{16}{179} a^{11} - \frac{5}{179} a^{10} - \frac{57}{179} a^{9} + \frac{14}{179} a^{8} + \frac{88}{179} a^{7} - \frac{73}{179} a^{6} + \frac{30}{179} a^{5} - \frac{55}{179} a^{4} + \frac{69}{179} a^{3} - \frac{43}{179} a^{2} + \frac{33}{179} a + \frac{27}{179}$, $\frac{1}{44584783} a^{41} - \frac{79331}{44584783} a^{40} + \frac{120426}{44584783} a^{39} - \frac{56178}{44584783} a^{38} - \frac{12998743}{44584783} a^{37} - \frac{4993110}{44584783} a^{36} - \frac{13595346}{44584783} a^{35} - \frac{3268423}{44584783} a^{34} - \frac{7765468}{44584783} a^{33} - \frac{6751833}{44584783} a^{32} - \frac{1606256}{44584783} a^{31} + \frac{8423877}{44584783} a^{30} + \frac{12158722}{44584783} a^{29} + \frac{10313602}{44584783} a^{28} - \frac{10726491}{44584783} a^{27} - \frac{19603559}{44584783} a^{26} - \frac{16165303}{44584783} a^{25} + \frac{10500887}{44584783} a^{24} - \frac{16156856}{44584783} a^{23} - \frac{12664245}{44584783} a^{22} - \frac{17213293}{44584783} a^{21} + \frac{12188922}{44584783} a^{20} + \frac{1708192}{44584783} a^{19} + \frac{2449640}{44584783} a^{18} - \frac{9591032}{44584783} a^{17} + \frac{6497}{44584783} a^{16} + \frac{495408}{44584783} a^{15} + \frac{6164152}{44584783} a^{14} + \frac{13299161}{44584783} a^{13} - \frac{7724255}{44584783} a^{12} + \frac{21747653}{44584783} a^{11} + \frac{14048049}{44584783} a^{10} + \frac{6302975}{44584783} a^{9} + \frac{10187438}{44584783} a^{8} + \frac{81386}{249077} a^{7} - \frac{16053468}{44584783} a^{6} + \frac{22133691}{44584783} a^{5} + \frac{14006446}{44584783} a^{4} + \frac{20290377}{44584783} a^{3} + \frac{18114497}{44584783} a^{2} - \frac{21985514}{44584783} a + \frac{16673536}{44584783}$, $\frac{1}{713137215452423} a^{42} - \frac{7661074}{713137215452423} a^{41} + \frac{846438090937}{713137215452423} a^{40} + \frac{736764066663}{713137215452423} a^{39} - \frac{1336688695915}{713137215452423} a^{38} - \frac{271666641505040}{713137215452423} a^{37} - \frac{176918309180665}{713137215452423} a^{36} + \frac{210202469578433}{713137215452423} a^{35} - \frac{272273272043377}{713137215452423} a^{34} + \frac{264357471954782}{713137215452423} a^{33} + \frac{21811035011433}{713137215452423} a^{32} - \frac{83739465977495}{713137215452423} a^{31} + \frac{298513681399637}{713137215452423} a^{30} + \frac{218039355148153}{713137215452423} a^{29} - \frac{72574156634696}{713137215452423} a^{28} + \frac{17537367116411}{713137215452423} a^{27} + \frac{229284790379740}{713137215452423} a^{26} + \frac{133333095091348}{713137215452423} a^{25} - \frac{189448319036215}{713137215452423} a^{24} - \frac{218635469112457}{713137215452423} a^{23} - \frac{117571537394431}{713137215452423} a^{22} + \frac{65939842132560}{713137215452423} a^{21} - \frac{173626213310051}{713137215452423} a^{20} + \frac{270088104960685}{713137215452423} a^{19} - \frac{10502856229368}{713137215452423} a^{18} - \frac{234848510943493}{713137215452423} a^{17} - \frac{177922375241404}{713137215452423} a^{16} - \frac{166618651811567}{713137215452423} a^{15} + \frac{351305906238201}{713137215452423} a^{14} + \frac{77946527290229}{713137215452423} a^{13} + \frac{37833220484599}{713137215452423} a^{12} + \frac{57877312867455}{713137215452423} a^{11} + \frac{84543277163859}{713137215452423} a^{10} + \frac{62507080680728}{713137215452423} a^{9} - \frac{69177779221382}{713137215452423} a^{8} - \frac{302937258844845}{713137215452423} a^{7} - \frac{263353923826093}{713137215452423} a^{6} - \frac{167895378112273}{713137215452423} a^{5} + \frac{220148712778010}{713137215452423} a^{4} - \frac{228599237937500}{713137215452423} a^{3} - \frac{333721693413459}{713137215452423} a^{2} + \frac{109405514909733}{713137215452423} a + \frac{109626420583699}{713137215452423}$, $\frac{1}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{43} + \frac{835279061664351438346317110071637267607586386489267807051342189685249565746113306107301612420860364170990947911393177590825452938938092866087038644469832684689476126314442601790586742610839525}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{42} - \frac{17343134698369069836575605471270847117002251840655030400859510651513983427842957852965567918550089841160831303881107505887452350971434555128409677192290578715633599800324254219889189699671403179133173}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{41} + \frac{7046372535009353297161385271147557657333590095841202490406450221329888073315059866160996231793507124507036946135916745053755334283180584615004811846259922277220118666777065092727366935333149976164191993576}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{40} - \frac{3696065890280467737970770269789705242644533619206743574132822363420725083260823819858910705302599770798107032526263465961392549146372355706450237867394583788489743874548657540398704006553567836542953507491}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{39} - \frac{974077802003694348968178270799229079413663461803378727869685560979113458616331750834807434329795124262795125828008480732229044756738508328733055256950063455096792870982876315642165807315030750974653504954}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{38} + \frac{259192018757667692917703140061374922396780510446221210197983270668557888541619333707560959145815787759644389851104043376969924221175708612126904893211200866881786789544181085267925545963070577587382456954273}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{37} + \frac{63192979480856349325748878519229779764215786476458443676650686632021529062305405314141339208428449310809906071539455390322812374573554764187399721283276182146091950520090451578780089329787151246252659787761}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{36} + \frac{819915984586567972000630637499059644986326756659379086151869567449585678468525138076041623974272771868212654352674799560673733102849604124643470277436390214014222644394839711981598814493176814817400208241686}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{35} + \frac{678890366925351177196754810220285631378472168351478397268934548645640516911816125156253521500823199806935574656219323742656084020309796879155895067024119270506226945591649461749000998926565117095910878890008}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{34} + \frac{123081376981545648821263265008373721728889143221651274095165200955533493174375719301688973681529306138259428708369371708980797326632384532952590750792363677018663891166124352850112587646081533593988640340811}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{33} + \frac{1107165446334942749049991924203991388182764724123054748561424254273109675927603881104296593210247960575282927903892620323706208527850262527587550525231422034073454660586389621535963339114941956349198722080862}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{32} - \frac{807462750082427585350858595241972643406869932246693541746230211107723889280703425204825240212924986924247165922798984428772520300697798906706114516063832196753691165457049847316640586200182772389383343000734}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{31} - \frac{1123305403720860195792792291363331076544400349938015060013292598127586991324367490978243685002540898690849734303868791467835515910023902939536760752957206581968396096592395579348818874528438446574575801375588}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{30} + \frac{984256689266528710183144690509476318983616365779061422981100941658783214717793559164629831559318749800024261536795050457059356369128257491351887674213149312725609824263528046313543995568936918681992020619359}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{29} + \frac{729397221200547064438250876231125895579965902208445358020015966393866587509620624391622653712158160035729041296370038927035883863886526952104640557495575378461931929700997762560494639183607477113117369874198}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{28} - \frac{359395917943194569795112055694299483037175295705746334450352125360569981143459679355665871872922291244165714397205160195430214952756086597742703980657320843841660759996962436255449548743998623508719163265970}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{27} + \frac{1093588266186044937199934280776307542469107454783325581909313476539787804494514409339022624170118341641020960620319015320898426823874341162889582899996353879535803737868669893956818177706850688559796473727801}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{26} + \frac{1424508462517216586057855494080956340028459780603404570372450008342614181676680506247954544131143506522424023527608784223703066704037853180892760692389054543425423329708965827488730237485330464969762014510149}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{25} - \frac{780869928884451158337945434421911566223649539063869141836899448576516658448381050768631034565237544978474657283956270287810701755305030610407206989637735488855977107795226473826023182093262322940541796856790}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{24} + \frac{1068582616793209716719235411189865500881868830212650425184098787271188383272438882149300524441597042989942141516824813508984618320328965641352936056105025657723778490982417147216575957890346286951263405288105}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{23} + \frac{1130487148524472889771558175307918370178113575088287851550145932337557025548191632348373641789098524959335109185016983609801626784709857607635634807181761012157209438208268758873823774014646002158790561568362}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{22} - \frac{653517119872052378104502200272481102598320234404255142288103273675538318119180031784266986130892200911296843458900409977298246623705558463445980171938606238567943068460428411838471540507651010348230510920249}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{21} + \frac{319712792248788421525487855760251269877753045422570240966526131316425429443209351129478680408445435887720441698120421925000835495996227597255687359732239533896247650505241482261268261102217766521128025894035}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{20} - \frac{1490764747356711117374067192767788801959315455558512991668030764978375088990469259623414217301421705311791045567073094538957400835284191354519607100068234740550813620750850660283825435105530300192160177272703}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{19} + \frac{1003046332936209102069938461023392549041500915222677784334374269062851706594067633232028981518936008836693191611081127378660949097173830461682369552060622851225259501832132038406399749380618629915842137942645}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{18} - \frac{8231948458954460668072130725962313886768230135577583939586363401910014719300384482579811041476476862617687972246657823333485953349786802478987725782891029322006923803757460792340513480069773645359509997743}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{17} - \frac{1315596834244011378909726553550619086029219727717243958171724936525500045666286486075923259864125336029088065098816014770462659248474789692622583862224558999870014302149570923234395938073016012801661510219830}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{16} - \frac{288021158844406764475470077708829187992012210309819050722446082818238638793740366605822487818355107368719048339763965947728149421075230299108946958716785159249927676057054528936677744374749569949857259613578}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{15} - \frac{1167706662324310010859277166658429836717640130528788687784575003001303320127691746721731262948405853251059911106209258325301126400662552797537149763744687969244068261966714972738043589519094342863577949608936}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{14} + \frac{1544922992712762523959951885898935961455355561014378517575710383839424543413662562033263166076109721538398379332170209375324733417048074208077637105571520969884278255160451995903984949847555750278669100650794}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{13} - \frac{610303806971524825434630602064668125558497979470534685891335373668044500354383587951332449541605946600794816368343809037126545499516816812546257378861998591557048371446146591078137427290405284688385027593581}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{12} - \frac{919390948178294821382917164400142124244501214118103411360327572575766489429858513633828142181697828651463336170054295590258742200125493457277618078253167084443255540158085573235825711226132999362244902423150}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{11} + \frac{1212197253324409406181733174685370841864679682492737711105575858020252229408692908881569422624281113433674641101215855491134686620307353669333519720427617412493369520670971594198323290804311200671516647801770}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{10} + \frac{103318089803478114331482575700173145070283540613738498861262537368140812618422007617685259833610408136709605468827696206228077646922366223287029981510084078923029168445976367366554877706949382655521984606373}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{9} - \frac{1076376609309102635193173726889539321645207283682005398559920928615436568474102956204395912577524821564708328725791153380725204827089520160182886717722120132405749958018888520391443388305086034172276325696052}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{8} - \frac{944299504055890616674300534186336861177412600743492051690963603674032089720193083298993052006013776892748592967381569216990409427732515587997920053724941687633164481295240965241354070129178461926250464684352}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{7} + \frac{573401994322135481507601336800321439529792361932209565917276141002220963025547192280717059106793048125857649101951253512872510437045809638624550820068245815559870831678922037237267965287905363228438664759474}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{6} - \frac{837783991056697557851079140220280868146177574944657345853976624038101102617298749831232954034968520753678301818633519696073168071161687505102565318067658678093986620162813921841491904985595446390779566117375}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{5} - \frac{1261493413102324926493182360598786974690138599098572772308195146703142562350036395781005576315744579940698562277289406374980469165614069795577199143766900617473928186764367541444296204054682963837989619808737}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{4} + \frac{29367931107750605397253934602724306071454774392750356008278253578641894931523978936461007971519839486646111924069979176021321651639062144464927352425007219278812478523936364049888160097142270516956404706537}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{3} + \frac{208949306229694307296795527367320702243202166124227492295264343668788925852401308566873981864569492530615065206516094381038988054460076435364257701421817078239994125140443859210470948503836499000503896706259}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a^{2} + \frac{1243542741109543214980064941246741636221779939459995476686405348777737852408122594492079878900400024416231898366842838414095269612583903797475959037688614154558745803453802894889615712989133592070005281152731}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251} a + \frac{1221093134634672375156658757666395071312462646709120021568050916531414411670406682247349059414862226813429772545992379765633833511606723506353493821264739334337024695512986256823829447811268030225906750256796}{3092469455330216994268791858556517727349216456763237973423784538850525692998810692250799443842870445430388439661038247396056771949343741898038659941602469649434686849515121934791662499091511680174252195284251}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $43$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{44}$ (as 44T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 44
The 44 conjugacy class representatives for $C_{44}$
Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.17624225.2, 11.11.31181719929966183601.1, 22.22.86534669543385676516186776267386878120889.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $22^{2}$ $44$ R $44$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ $44$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{4}$ $44$ $44$ $44$ $44$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{11}$ $44$ $44$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{4}$ $44$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
89Data not computed