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Properties

Label 44.0.56917636750...8125.1
Degree $44$
Signature $[0, 22]$
Discriminant $3^{22}\cdot 5^{33}\cdot 23^{42}$
Root discriminant $115.51$
Ramified primes $3, 5, 23$
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![5289377620231, 4334517932109, 9623895551811, 5934486807814, 15558382324711, 1571234349904, 17129615889050, -4127292620568, 13002314190842, -5657518814231, 7344731833131, -3964425484884, 3380011840722, -1995509837914, 1383546745188, -842528728138, 538767595505, -328053581132, 206763685713, -122773157599, 78732114704, -44059838242, 29302447717, -14753393126, 10313292941, -4440100185, 3284745340, -1155354845, 904617380, -250737465, 206824476, -43912989, 37901220, -6011769, 5390254, -621515, 575057, -46458, 44104, -2354, 2282, -72, 71, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 + 71*x^42 - 72*x^41 + 2282*x^40 - 2354*x^39 + 44104*x^38 - 46458*x^37 + 575057*x^36 - 621515*x^35 + 5390254*x^34 - 6011769*x^33 + 37901220*x^32 - 43912989*x^31 + 206824476*x^30 - 250737465*x^29 + 904617380*x^28 - 1155354845*x^27 + 3284745340*x^26 - 4440100185*x^25 + 10313292941*x^24 - 14753393126*x^23 + 29302447717*x^22 - 44059838242*x^21 + 78732114704*x^20 - 122773157599*x^19 + 206763685713*x^18 - 328053581132*x^17 + 538767595505*x^16 - 842528728138*x^15 + 1383546745188*x^14 - 1995509837914*x^13 + 3380011840722*x^12 - 3964425484884*x^11 + 7344731833131*x^10 - 5657518814231*x^9 + 13002314190842*x^8 - 4127292620568*x^7 + 17129615889050*x^6 + 1571234349904*x^5 + 15558382324711*x^4 + 5934486807814*x^3 + 9623895551811*x^2 + 4334517932109*x + 5289377620231)
 
gp: K = bnfinit(x^44 - x^43 + 71*x^42 - 72*x^41 + 2282*x^40 - 2354*x^39 + 44104*x^38 - 46458*x^37 + 575057*x^36 - 621515*x^35 + 5390254*x^34 - 6011769*x^33 + 37901220*x^32 - 43912989*x^31 + 206824476*x^30 - 250737465*x^29 + 904617380*x^28 - 1155354845*x^27 + 3284745340*x^26 - 4440100185*x^25 + 10313292941*x^24 - 14753393126*x^23 + 29302447717*x^22 - 44059838242*x^21 + 78732114704*x^20 - 122773157599*x^19 + 206763685713*x^18 - 328053581132*x^17 + 538767595505*x^16 - 842528728138*x^15 + 1383546745188*x^14 - 1995509837914*x^13 + 3380011840722*x^12 - 3964425484884*x^11 + 7344731833131*x^10 - 5657518814231*x^9 + 13002314190842*x^8 - 4127292620568*x^7 + 17129615889050*x^6 + 1571234349904*x^5 + 15558382324711*x^4 + 5934486807814*x^3 + 9623895551811*x^2 + 4334517932109*x + 5289377620231, 1)
 

Normalized defining polynomial

\( x^{44} - x^{43} + 71 x^{42} - 72 x^{41} + 2282 x^{40} - 2354 x^{39} + 44104 x^{38} - 46458 x^{37} + 575057 x^{36} - 621515 x^{35} + 5390254 x^{34} - 6011769 x^{33} + 37901220 x^{32} - 43912989 x^{31} + 206824476 x^{30} - 250737465 x^{29} + 904617380 x^{28} - 1155354845 x^{27} + 3284745340 x^{26} - 4440100185 x^{25} + 10313292941 x^{24} - 14753393126 x^{23} + 29302447717 x^{22} - 44059838242 x^{21} + 78732114704 x^{20} - 122773157599 x^{19} + 206763685713 x^{18} - 328053581132 x^{17} + 538767595505 x^{16} - 842528728138 x^{15} + 1383546745188 x^{14} - 1995509837914 x^{13} + 3380011840722 x^{12} - 3964425484884 x^{11} + 7344731833131 x^{10} - 5657518814231 x^{9} + 13002314190842 x^{8} - 4127292620568 x^{7} + 17129615889050 x^{6} + 1571234349904 x^{5} + 15558382324711 x^{4} + 5934486807814 x^{3} + 9623895551811 x^{2} + 4334517932109 x + 5289377620231 \)

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

Invariants

Degree:  $44$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 22]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5691763675094128540646263027975260328820026154733783913534198355344240553677082061767578125=3^{22}\cdot 5^{33}\cdot 23^{42}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.51$
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:  $3, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(345=3\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{345}(256,·)$, $\chi_{345}(1,·)$, $\chi_{345}(259,·)$, $\chi_{345}(4,·)$, $\chi_{345}(263,·)$, $\chi_{345}(137,·)$, $\chi_{345}(139,·)$, $\chi_{345}(271,·)$, $\chi_{345}(16,·)$, $\chi_{345}(17,·)$, $\chi_{345}(151,·)$, $\chi_{345}(152,·)$, $\chi_{345}(68,·)$, $\chi_{345}(154,·)$, $\chi_{345}(158,·)$, $\chi_{345}(31,·)$, $\chi_{345}(289,·)$, $\chi_{345}(293,·)$, $\chi_{345}(38,·)$, $\chi_{345}(169,·)$, $\chi_{345}(301,·)$, $\chi_{345}(49,·)$, $\chi_{345}(53,·)$, $\chi_{345}(182,·)$, $\chi_{345}(287,·)$, $\chi_{345}(64,·)$, $\chi_{345}(331,·)$, $\chi_{345}(196,·)$, $\chi_{345}(203,·)$, $\chi_{345}(332,·)$, $\chi_{345}(334,·)$, $\chi_{345}(83,·)$, $\chi_{345}(212,·)$, $\chi_{345}(218,·)$, $\chi_{345}(143,·)$, $\chi_{345}(94,·)$, $\chi_{345}(272,·)$, $\chi_{345}(227,·)$, $\chi_{345}(107,·)$, $\chi_{345}(113,·)$, $\chi_{345}(211,·)$, $\chi_{345}(121,·)$, $\chi_{345}(122,·)$, $\chi_{345}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{788748162049} a^{34} - \frac{47140934156}{788748162049} a^{33} + \frac{309627652424}{788748162049} a^{32} - \frac{50699479007}{788748162049} a^{31} + \frac{295993578030}{788748162049} a^{30} - \frac{375286130378}{788748162049} a^{29} - \frac{966550082}{788748162049} a^{28} + \frac{343746287882}{788748162049} a^{27} - \frac{54339769711}{788748162049} a^{26} - \frac{98152387442}{788748162049} a^{25} + \frac{238406873608}{788748162049} a^{24} - \frac{156915694908}{788748162049} a^{23} + \frac{299364438516}{788748162049} a^{22} - \frac{85164682969}{788748162049} a^{21} - \frac{95042883984}{788748162049} a^{20} - \frac{318910272560}{788748162049} a^{19} - \frac{367003498427}{788748162049} a^{18} + \frac{100879386855}{788748162049} a^{17} - \frac{304996279191}{788748162049} a^{16} + \frac{141714670909}{788748162049} a^{15} + \frac{164152364977}{788748162049} a^{14} - \frac{20053825540}{788748162049} a^{13} - \frac{306247465325}{788748162049} a^{12} - \frac{17365054627}{788748162049} a^{11} + \frac{208106703761}{788748162049} a^{10} - \frac{161275377022}{788748162049} a^{9} - \frac{225484062111}{788748162049} a^{8} + \frac{110032384593}{788748162049} a^{7} + \frac{59297751020}{788748162049} a^{6} - \frac{339171063562}{788748162049} a^{5} + \frac{142703379833}{788748162049} a^{4} - \frac{4729698341}{788748162049} a^{3} - \frac{101718421595}{788748162049} a^{2} - \frac{372367917072}{788748162049} a - \frac{259685920631}{788748162049}$, $\frac{1}{788748162049} a^{35} + \frac{50401955370}{788748162049} a^{33} - \frac{205485042739}{788748162049} a^{32} - \frac{313368995030}{788748162049} a^{31} - \frac{133199845185}{788748162049} a^{30} - \frac{79160489821}{788748162049} a^{29} - \frac{90525379441}{788748162049} a^{28} + \frac{89184173923}{788748162049} a^{27} - \frac{289854198913}{788748162049} a^{26} - \frac{336013416877}{788748162049} a^{25} - \frac{288459874069}{788748162049} a^{24} + \frac{354641181592}{788748162049} a^{23} + \frac{296405272451}{788748162049} a^{22} - \frac{134071590167}{788748162049} a^{21} + \frac{223342477746}{788748162049} a^{20} + \frac{20435185417}{788748162049} a^{19} - \frac{319186904475}{788748162049} a^{18} + \frac{358891765633}{788748162049} a^{17} - \frac{280824756967}{788748162049} a^{16} - \frac{221239924878}{788748162049} a^{15} - \frac{224745009118}{788748162049} a^{14} - \frac{286967800}{788748162049} a^{13} - \frac{51750004718}{788748162049} a^{12} - \frac{318435269518}{788748162049} a^{11} + \frac{275074844392}{788748162049} a^{10} - \frac{337745719910}{788748162049} a^{9} + \frac{56027069507}{788748162049} a^{8} - \frac{87540397868}{788748162049} a^{7} - \frac{343135277082}{788748162049} a^{6} - \frac{156679011677}{788748162049} a^{5} - \frac{326290813308}{788748162049} a^{4} - \frac{24858078709}{788748162049} a^{3} - \frac{58322243638}{788748162049} a^{2} + \frac{362002945664}{788748162049} a - \frac{387738015398}{788748162049}$, $\frac{1}{788748162049} a^{36} + \frac{124710740321}{788748162049} a^{33} + \frac{333239944079}{788748162049} a^{32} + \frac{335645564437}{788748162049} a^{31} + \frac{122516235906}{788748162049} a^{30} + \frac{121981788070}{788748162049} a^{29} - \frac{136305751625}{788748162049} a^{28} + \frac{270242509181}{788748162049} a^{27} - \frac{329350517374}{788748162049} a^{26} - \frac{27141778225}{788748162049} a^{25} + \frac{387249126422}{788748162049} a^{24} - \frac{84375935423}{788748162049} a^{23} + \frac{290349157504}{788748162049} a^{22} - \frac{115264866242}{788748162049} a^{21} - \frac{162687904856}{788748162049} a^{20} + \frac{89000168316}{788748162049} a^{19} + \frac{50888850840}{788748162049} a^{18} - \frac{103842902161}{788748162049} a^{17} - \frac{56506717713}{788748162049} a^{16} + \frac{275816902086}{788748162049} a^{15} - \frac{13105510719}{788748162049} a^{14} + \frac{17809331132}{788748162049} a^{13} + \frac{221383000674}{788748162049} a^{12} + \frac{63631100340}{788748162049} a^{11} - \frac{280767460626}{788748162049} a^{10} - \frac{337243755561}{788748162049} a^{9} + \frac{184105952428}{788748162049} a^{8} + \frac{353043024859}{788748162049} a^{7} + \frac{228738304726}{788748162049} a^{6} - \frac{64658419485}{788748162049} a^{5} + \frac{203164371327}{788748162049} a^{4} + \frac{357265426589}{788748162049} a^{3} + \frac{286726357586}{788748162049} a^{2} - \frac{161841017594}{788748162049} a + \frac{96659991605}{788748162049}$, $\frac{1}{788748162049} a^{37} - \frac{351397598702}{788748162049} a^{33} + \frac{308936706405}{788748162049} a^{32} + \frac{113789856600}{788748162049} a^{31} - \frac{41815120967}{788748162049} a^{30} + \frac{243895682850}{788748162049} a^{29} - \frac{103041543146}{788748162049} a^{28} + \frac{245605584077}{788748162049} a^{27} - \frac{72350083272}{788748162049} a^{26} + \frac{259606163246}{788748162049} a^{25} - \frac{314351167499}{788748162049} a^{24} + \frac{332808386508}{788748162049} a^{23} - \frac{363841832144}{788748162049} a^{22} + \frac{168892202761}{788748162049} a^{21} - \frac{379926241291}{788748162049} a^{20} - \frac{17606569690}{788748162049} a^{19} + \frac{341785921935}{788748162049} a^{18} - \frac{150768216551}{788748162049} a^{17} - \frac{232903494230}{788748162049} a^{16} - \frac{4098439358}{788748162049} a^{15} + \frac{254237110349}{788748162049} a^{14} + \frac{203368072432}{788748162049} a^{13} + \frac{248304320682}{788748162049} a^{12} + \frac{146222656361}{788748162049} a^{11} - \frac{362832066411}{788748162049} a^{10} - \frac{5017604560}{788748162049} a^{9} + \frac{197650704649}{788748162049} a^{8} + \frac{205044288696}{788748162049} a^{7} + \frac{99200826882}{788748162049} a^{6} + \frac{294493124980}{788748162049} a^{5} - \frac{199083396438}{788748162049} a^{4} + \frac{251506669806}{788748162049} a^{3} - \frac{141503756752}{788748162049} a^{2} - \frac{9228451994}{788748162049} a - \frac{28424175407}{788748162049}$, $\frac{1}{788748162049} a^{38} - \frac{149082002623}{788748162049} a^{33} + \frac{26647274837}{788748162049} a^{32} + \frac{148010532835}{788748162049} a^{31} + \frac{245854081243}{788748162049} a^{30} - \frac{153243095958}{788748162049} a^{29} + \frac{81765501539}{788748162049} a^{28} + \frac{177271549997}{788748162049} a^{27} + \frac{129430586379}{788748162049} a^{26} - \frac{30442260546}{788748162049} a^{25} - \frac{233027253721}{788748162049} a^{24} + \frac{302228577838}{788748162049} a^{23} - \frac{266209172040}{788748162049} a^{22} + \frac{290700658479}{788748162049} a^{21} + \frac{250123300796}{788748162049} a^{20} + \frac{365885693875}{788748162049} a^{19} + \frac{308720918104}{788748162049} a^{18} - \frac{66022942584}{788748162049} a^{17} - \frac{11597219368}{788748162049} a^{16} + \frac{134815510166}{788748162049} a^{15} - \frac{241560779564}{788748162049} a^{14} + \frac{361351846838}{788748162049} a^{13} - \frac{364550016016}{788748162049} a^{12} - \frac{272410165201}{788748162049} a^{11} + \frac{282548668557}{788748162049} a^{10} + \frac{57776468882}{788748162049} a^{9} - \frac{110316080776}{788748162049} a^{8} - \frac{281177086064}{788748162049} a^{7} + \frac{189975837586}{788748162049} a^{6} - \frac{100254175780}{788748162049} a^{5} - \frac{283406303401}{788748162049} a^{4} + \frac{209410524036}{788748162049} a^{3} + \frac{315048346254}{788748162049} a^{2} - \frac{11031498699}{788748162049} a + \frac{47560203061}{788748162049}$, $\frac{1}{788748162049} a^{39} - \frac{365176701450}{788748162049} a^{33} + \frac{304185396973}{788748162049} a^{32} - \frac{60849580144}{788748162049} a^{31} + \frac{377955395900}{788748162049} a^{30} + \frac{285314629378}{788748162049} a^{29} + \frac{74768069607}{788748162049} a^{28} + \frac{222173687510}{788748162049} a^{27} + \frac{126189108590}{788748162049} a^{26} - \frac{375604422703}{788748162049} a^{25} + \frac{328876332917}{788748162049} a^{24} - \frac{304827302768}{788748162049} a^{23} - \frac{346137828848}{788748162049} a^{22} - \frac{3421415502}{788748162049} a^{21} - \frac{123360073615}{788748162049} a^{20} - \frac{117371704329}{788748162049} a^{19} + \frac{262130899125}{788748162049} a^{18} - \frac{278579714614}{788748162049} a^{17} + \frac{381954858325}{788748162049} a^{16} - \frac{97614757591}{788748162049} a^{15} + \frac{379088286750}{788748162049} a^{14} - \frac{285444893067}{788748162049} a^{13} - \frac{375397141295}{788748162049} a^{12} - \frac{299233487675}{788748162049} a^{11} - \frac{216076435494}{788748162049} a^{10} + \frac{105436658769}{788748162049} a^{9} + \frac{358358266062}{788748162049} a^{8} - \frac{233108495065}{788748162049} a^{7} - \frac{259358297183}{788748162049} a^{6} + \frac{210042868059}{788748162049} a^{5} + \frac{253598838078}{788748162049} a^{4} + \frac{318037933055}{788748162049} a^{3} + \frac{139373290513}{788748162049} a^{2} + \frac{353154154021}{788748162049} a - \frac{186749673308}{788748162049}$, $\frac{1}{788748162049} a^{40} + \frac{208988277619}{788748162049} a^{33} - \frac{389037297785}{788748162049} a^{32} - \frac{127054124521}{788748162049} a^{31} - \frac{48713251635}{788748162049} a^{30} - \frac{186796604504}{788748162049} a^{29} - \frac{281853908192}{788748162049} a^{28} - \frac{102293254496}{788748162049} a^{27} + \frac{33047311647}{788748162049} a^{26} + \frac{232502371618}{788748162049} a^{25} - \frac{381349827565}{788748162049} a^{24} - \frac{356837343707}{788748162049} a^{23} + \frac{367404634800}{788748162049} a^{22} + \frac{245948625409}{788748162049} a^{21} + \frac{350052089572}{788748162049} a^{20} - \frac{112170160452}{788748162049} a^{19} - \frac{338017692770}{788748162049} a^{18} + \frac{108735851295}{788748162049} a^{17} - \frac{238228011622}{788748162049} a^{16} + \frac{293768973282}{788748162049} a^{15} - \frac{37733669990}{788748162049} a^{14} - \frac{184029606961}{788748162049} a^{13} - \frac{172516616858}{788748162049} a^{12} - \frac{90820553021}{788748162049} a^{11} + \frac{368512405382}{788748162049} a^{10} - \frac{92432381854}{788748162049} a^{9} + \frac{354902848584}{788748162049} a^{8} - \frac{5322093064}{788748162049} a^{7} + \frac{309377300803}{788748162049} a^{6} + \frac{309614472270}{788748162049} a^{5} - \frac{27908682651}{788748162049} a^{4} + \frac{239062227618}{788748162049} a^{3} + \frac{32252361188}{788748162049} a^{2} - \frac{190571159737}{788748162049} a + \frac{144795943962}{788748162049}$, $\frac{1}{724859560923031} a^{41} - \frac{345}{724859560923031} a^{40} - \frac{89}{724859560923031} a^{39} - \frac{195}{724859560923031} a^{38} + \frac{176}{724859560923031} a^{37} - \frac{321}{724859560923031} a^{36} + \frac{315}{724859560923031} a^{35} - \frac{162}{724859560923031} a^{34} + \frac{14216266741582}{724859560923031} a^{33} + \frac{82605178028716}{724859560923031} a^{32} + \frac{273458636437372}{724859560923031} a^{31} - \frac{13295959459359}{724859560923031} a^{30} + \frac{36772142921572}{724859560923031} a^{29} - \frac{47281481137050}{724859560923031} a^{28} + \frac{243902683807282}{724859560923031} a^{27} - \frac{148473546022569}{724859560923031} a^{26} - \frac{361594747643532}{724859560923031} a^{25} + \frac{141425164267399}{724859560923031} a^{24} - \frac{236020020705387}{724859560923031} a^{23} - \frac{199780801839337}{724859560923031} a^{22} + \frac{166074326459738}{724859560923031} a^{21} - \frac{138457025724936}{724859560923031} a^{20} - \frac{34164178307512}{724859560923031} a^{19} + \frac{208450110240454}{724859560923031} a^{18} - \frac{272635453932800}{724859560923031} a^{17} - \frac{127754016082495}{724859560923031} a^{16} - \frac{224980667588927}{724859560923031} a^{15} - \frac{134713562797118}{724859560923031} a^{14} + \frac{270690447549907}{724859560923031} a^{13} + \frac{87175468306403}{724859560923031} a^{12} + \frac{80956269482952}{724859560923031} a^{11} - \frac{51521449115575}{724859560923031} a^{10} - \frac{347703750535864}{724859560923031} a^{9} - \frac{59566032343386}{724859560923031} a^{8} + \frac{130335238501742}{724859560923031} a^{7} - \frac{183945154082088}{724859560923031} a^{6} - \frac{269155452784265}{724859560923031} a^{5} - \frac{269179629949368}{724859560923031} a^{4} - \frac{305960263728412}{724859560923031} a^{3} - \frac{94481386601946}{724859560923031} a^{2} - \frac{244249096751468}{724859560923031} a + \frac{40919093061435}{724859560923031}$, $\frac{1}{99429710831373085301} a^{42} + \frac{42774}{99429710831373085301} a^{41} + \frac{33742632}{99429710831373085301} a^{40} + \frac{40107875}{99429710831373085301} a^{39} - \frac{40607951}{99429710831373085301} a^{38} - \frac{52670207}{99429710831373085301} a^{37} + \frac{18697230}{99429710831373085301} a^{36} + \frac{32388739}{99429710831373085301} a^{35} + \frac{20952746}{99429710831373085301} a^{34} - \frac{48257833110889189666}{99429710831373085301} a^{33} + \frac{25603365012494119160}{99429710831373085301} a^{32} - \frac{48071769804630103926}{99429710831373085301} a^{31} + \frac{25622305239300607669}{99429710831373085301} a^{30} + \frac{33908505098025318501}{99429710831373085301} a^{29} + \frac{14499611697803587689}{99429710831373085301} a^{28} - \frac{45962135125067037592}{99429710831373085301} a^{27} + \frac{8280551178646563118}{99429710831373085301} a^{26} - \frac{36969101189509117898}{99429710831373085301} a^{25} + \frac{27816235698751421776}{99429710831373085301} a^{24} + \frac{25557708087079778184}{99429710831373085301} a^{23} - \frac{33659699728713485870}{99429710831373085301} a^{22} - \frac{14341291554148329073}{99429710831373085301} a^{21} + \frac{45630332838815402184}{99429710831373085301} a^{20} - \frac{38946805599226072079}{99429710831373085301} a^{19} - \frac{48873529168374383949}{99429710831373085301} a^{18} + \frac{29893991014529054217}{99429710831373085301} a^{17} + \frac{20132304974725572031}{99429710831373085301} a^{16} + \frac{31000681240436568972}{99429710831373085301} a^{15} + \frac{38172882027787767217}{99429710831373085301} a^{14} - \frac{9760097172802192412}{99429710831373085301} a^{13} - \frac{29874435352135626151}{99429710831373085301} a^{12} - \frac{48412402866534383832}{99429710831373085301} a^{11} + \frac{49577156134506087375}{99429710831373085301} a^{10} + \frac{47074234848683322710}{99429710831373085301} a^{9} - \frac{6305819191230141276}{99429710831373085301} a^{8} + \frac{367341233298325893}{99429710831373085301} a^{7} - \frac{22322852452947180222}{99429710831373085301} a^{6} - \frac{1556003166976609092}{99429710831373085301} a^{5} + \frac{47926265029149141627}{99429710831373085301} a^{4} - \frac{21174123902930272958}{99429710831373085301} a^{3} - \frac{17918446987675193895}{99429710831373085301} a^{2} - \frac{33092867711426914189}{99429710831373085301} a - \frac{43770600137692535770}{99429710831373085301}$, $\frac{1}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{43} - \frac{365082167879589809799995438676328564955580413259322984844953555538752781707419194730942617290145536186562433660249184319775079401848270464906326265}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{42} + \frac{446978898121213194848029692459538657989056140133634394751757704584934362972247708126433420047529254578420440129486885126615234931023957830527883393455743}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{41} - \frac{1318050848967662558643964290937309691260813365840768495176742560470205233209561848302795160188361701881081470747225689881733227486051841700965726517358403646}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{40} + \frac{6979398641734067215843302186184788473405951561530341896491459737900186029024658593596159933660873874178055063694161084017817036876071185438642175703905468404}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{39} - \frac{1902664602885130142628451635909151838186569208832131254669455263584585216594041938466154024715529584481650859411354010677789598625599134081970672863139887982}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{38} + \frac{2577540845943329780147856960256204448862669778957147299934607926775528567544748525255804326087108009908134304954254240613488416053397935870908439762637555697}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{37} - \frac{4400682060021866382628318974900774252540436266326966260790277149908988936925234104908954104033039644953659242037575589550731409696253036827979551208782893726}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{36} + \frac{5039272761022001314117136610193153464035517819405693076205310602331497499836897157782401129623391384219134155860527520525350286615600741166813832804220654578}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{35} - \frac{6477477884933711971711750159728903469215632842615456116734384391233790901173409674487111794991451198403248317138075432716442922612294685369295353657710425476}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{34} + \frac{5573757218079936926294932459367660411458711211563984290069896525146849045129662648609395128515198624553522200320377424276883015489075550201789783180694321931418165779194}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{33} - \frac{3041280067548060411895918743328225951332271011885801587015623973813508242117051129546247367342543988273985402320622409505444341666231745431917541087149043500755469086079}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{32} - \frac{2127041256575855706010435986258826178557351453834729476397994081584067847535363070933036147757454250113874385008648465404283995588140294318416138842560213894128677233343}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{31} - \frac{5622307566078693417090130247359339835721976966879391492669130222274936115087123734029967976642150351757950683949414775049479347503498588694128548611110847420950229131866}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{30} - \frac{3301040991596856884548291230697915005151861916830415637081993225810911753793972798847419550173723019105566714117958176082930214818320179093471549261864897620039988623057}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{29} - \frac{1752435107199461807285631697400739942639843153578655274031856702819457813990700625236761100419293486688607586171695780384044625643784278525069840799586916975707170804521}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{28} - \frac{470157988267748911411886447227753337848963052920528623702029682787368143065912778837720832401158382196586378654582549374561779072416848675992653380921894194272927208434}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{27} + \frac{2029346580766576281718893774796845651782259858723912101592483968726234320158025358407367367666725571919702143921763185052189309767232264613124260026013836111500700227562}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{26} + \frac{3073309380257503639436376819272574709843398047319299795407027938653633993363261572830180328341340195467519327053472138220816676593008145166312572636430338423237743842439}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{25} - \frac{5924552865453933625452675589744055768607121717172086975393183118307145573414222439767309659309917520004038693664285898206763796782837384434717443022173406815970762892344}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{24} + \frac{2479014237409767477909381273364611988925139291337989639774985931381911001754116005843500572565788540240826698930073482775736742482917475111838074714236652502928382547851}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{23} + \frac{5940473173247815585788593244188294745799141409366006892475932952884884449023798003267039474074097038272580815147980957386125047710591020555812383325539287232931423540073}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{22} - \frac{220885889859492575755105784585212542816618208274435717400538758017081416700016627700937490728159496294160919244605109616216452912042684316928664399597846975664769805876}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{21} - \frac{1105591797146742301389002205400589915616019670117157475710081109014019504091948341040810546582416524090355569954602284366301879480778425327626111286547067637663948995855}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{20} + \frac{2642179688878813462646657253766195879401643823760804414469585476654220494652196201937360219452761550812920572327211454361451305987714330610333685835340942247340661598192}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{19} + \frac{4377387805246520052102992441317834068239772014501774602128774933993340153544526081557573886129434968339344448694498626661126888512270376514980915969108098771630143044557}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{18} - \frac{1309090207090208112392893943053449622523641759332480812238693056244919317738351193130534942663843801086091779971087841261562879272794571432448373269842093243155008962765}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{17} - \frac{3072917148353307861485777605177264216608339073678437961262246656594811712535808173802908703932622522560455851954136407780284262736485002834347458576221761568002563461067}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{16} + \frac{4139226233830257515504527417060155752541461013030973368448290808996741197575270192567728960627104299317913529039385026984077231316088327803631717912381802437053372829345}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{15} - \frac{3342727222398231673783367479229310962488189320150144316905289871623795281717087471450220561583935161399365867705627811797693480611997529234425832505787411916313343630446}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{14} + \frac{3309010847156440081817704363332059811341399052775857959602134572960773083996054596235755874810545444285446198684381955044299615396549664334738087520564873486522843781597}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{13} + \frac{2492092370604275528462923099494689735075641343146705885697866668294604467002661701801910909205002067501930474829743347350934381637396656039701733753400063191717979001282}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{12} + \frac{1888482095059083722727922373196015524212406916916546075317692267573034805934211948808056502193852214112579656860203525529100385032535007079478507195883034169885858074753}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{11} + \frac{4046881546733049687905063205108327974538702034565432117164973766715237657518417137534944569838689832283100725280115483813139396570079558995575944489446022204435949840181}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{10} + \frac{923649505444758158945968101840527470415486519309503688297611145431896993210781775471884003780008708535050326197507865740981609378874706917021388193665156971473954896582}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{9} - \frac{5135691335597227867181943087606122305369154489935631656302646033942734859405506381503559838659888805312137699955616102575449295754932605775013084199473012149253657487900}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{8} - \frac{6143759713788866056992745130749790781105377185305627038743385175733997251807728380095066076474040075516252783083390235164968996821682239926209023531321884283549682570596}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{7} - \frac{4244474779732813897384431859420469008401302277061466253749129730091462900386729899738582331443462015613183376110252451118535240666877798812715721079964879509192313136195}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{6} + \frac{3217850275563976250925911774983996023467454785124191456298600115857822531962452735405787178107646861799580041272774309452736327325381570136380707793996569100812242027128}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{5} - \frac{1252545394660634523425996825427112596290482529015569680399253210367000003155776082880343938872988583143594676489001239345584396323757683367662705236763848700172580504282}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{4} + \frac{3404732527589419567514136275330598862714196957756564741589611376412865589662764447631553493413068733959117738113230581898751231207544268849635356228573483935403319093808}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{3} + \frac{2188638010877483001814529241637908063196862100672951873016728188053777460796581138502034722948779348251208140640835549752349311089654777443500825199797132665404138201558}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a^{2} + \frac{4010934280920929441956595138618736440311916494902750507439645821242220005514414118761556788890259461339737135392307417393768686705669017061334007924175877319052174182183}{12502404839575580188506968565734666654799503208158551416140144090460299340901124164045443626958290337671162543471887122588429401191122446878634914170774804193005355002559} a + \frac{216282093159268153710931365050921497677268539576844131480619750624616682824197485224885709798498475321485625733190720729962371828253745731843405316548637169}{2363681653538203938356672567456863722979371566999278940527469970124472560876145387117029856252588076870339746647704582134336183529677880794466575390254142089}$

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:  $21$
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

\(\Q(\sqrt{5}) \), 4.0.595125.1, \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1

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 $44$ R R $44$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ $44$ $44$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{4}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $44$ $22^{2}$ $44$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{11}$ $44$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{4}$

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
3Data not computed
5Data not computed
23Data not computed