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Properties

Label 44.0.653...000.2
Degree $44$
Signature $[0, 22]$
Discriminant $6.535\times 10^{85}$
Root discriminant $89.20$
Ramified primes $2, 5, 23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 47*x^42 + 1036*x^40 - 14238*x^38 + 136828*x^36 - 977616*x^34 + 5391753*x^32 - 23533371*x^30 + 82732335*x^28 - 237396295*x^26 + 562190611*x^24 - 1110544651*x^22 + 1852435411*x^20 - 2651394691*x^18 + 3329083366*x^16 - 3775046236*x^14 + 3998027671*x^12 - 4080345361*x^10 + 4101904756*x^8 - 4105687106*x^6 + 4106094436*x^4 - 4106117712*x^2 + 4106118241)
 
gp: K = bnfinit(x^44 - 47*x^42 + 1036*x^40 - 14238*x^38 + 136828*x^36 - 977616*x^34 + 5391753*x^32 - 23533371*x^30 + 82732335*x^28 - 237396295*x^26 + 562190611*x^24 - 1110544651*x^22 + 1852435411*x^20 - 2651394691*x^18 + 3329083366*x^16 - 3775046236*x^14 + 3998027671*x^12 - 4080345361*x^10 + 4101904756*x^8 - 4105687106*x^6 + 4106094436*x^4 - 4106117712*x^2 + 4106118241, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4106118241, 0, -4106117712, 0, 4106094436, 0, -4105687106, 0, 4101904756, 0, -4080345361, 0, 3998027671, 0, -3775046236, 0, 3329083366, 0, -2651394691, 0, 1852435411, 0, -1110544651, 0, 562190611, 0, -237396295, 0, 82732335, 0, -23533371, 0, 5391753, 0, -977616, 0, 136828, 0, -14238, 0, 1036, 0, -47, 0, 1]);
 

\( x^{44} - 47 x^{42} + 1036 x^{40} - 14238 x^{38} + 136828 x^{36} - 977616 x^{34} + 5391753 x^{32} - 23533371 x^{30} + 82732335 x^{28} - 237396295 x^{26} + 562190611 x^{24} - 1110544651 x^{22} + 1852435411 x^{20} - 2651394691 x^{18} + 3329083366 x^{16} - 3775046236 x^{14} + 3998027671 x^{12} - 4080345361 x^{10} + 4101904756 x^{8} - 4105687106 x^{6} + 4106094436 x^{4} - 4106117712 x^{2} + 4106118241 \)

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(653\!\cdots\!000\)\(\medspace = 2^{44}\cdot 5^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $89.20$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 5, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $44$
This field is Galois and abelian over $\Q$.
Conductor:  \(460=2^{2}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{460}(1,·)$, $\chi_{460}(131,·)$, $\chi_{460}(261,·)$, $\chi_{460}(129,·)$, $\chi_{460}(141,·)$, $\chi_{460}(271,·)$, $\chi_{460}(19,·)$, $\chi_{460}(149,·)$, $\chi_{460}(151,·)$, $\chi_{460}(389,·)$, $\chi_{460}(419,·)$, $\chi_{460}(31,·)$, $\chi_{460}(41,·)$, $\chi_{460}(71,·)$, $\chi_{460}(301,·)$, $\chi_{460}(309,·)$, $\chi_{460}(311,·)$, $\chi_{460}(441,·)$, $\chi_{460}(159,·)$, $\chi_{460}(189,·)$, $\chi_{460}(319,·)$, $\chi_{460}(331,·)$, $\chi_{460}(211,·)$, $\chi_{460}(199,·)$, $\chi_{460}(329,·)$, $\chi_{460}(459,·)$, $\chi_{460}(79,·)$, $\chi_{460}(81,·)$, $\chi_{460}(339,·)$, $\chi_{460}(121,·)$, $\chi_{460}(429,·)$, $\chi_{460}(351,·)$, $\chi_{460}(99,·)$, $\chi_{460}(101,·)$, $\chi_{460}(231,·)$, $\chi_{460}(361,·)$, $\chi_{460}(359,·)$, $\chi_{460}(109,·)$, $\chi_{460}(229,·)$, $\chi_{460}(371,·)$, $\chi_{460}(89,·)$, $\chi_{460}(249,·)$, $\chi_{460}(379,·)$, $\chi_{460}(381,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{64079} a^{23} - \frac{23}{64079} a^{21} + \frac{230}{64079} a^{19} - \frac{1311}{64079} a^{17} + \frac{4692}{64079} a^{15} - \frac{10948}{64079} a^{13} + \frac{16744}{64079} a^{11} - \frac{16445}{64079} a^{9} + \frac{9867}{64079} a^{7} - \frac{3289}{64079} a^{5} + \frac{506}{64079} a^{3} - \frac{23}{64079} a$, $\frac{1}{1836311903} a^{24} - \frac{701408757}{1836311903} a^{22} + \frac{740497154}{1836311903} a^{20} + \frac{310525523}{1836311903} a^{18} - \frac{797202147}{1836311903} a^{16} + \frac{821033279}{1836311903} a^{14} - \frac{396952661}{1836311903} a^{12} + \frac{316277499}{1836311903} a^{10} - \frac{8768956}{1836311903} a^{8} - \frac{913769829}{1836311903} a^{6} + \frac{328469460}{1836311903} a^{4} - \frac{400109299}{1836311903} a^{2} - \frac{6765}{28657}$, $\frac{1}{1836311903} a^{25} - \frac{25}{1836311903} a^{23} - \frac{701408458}{1836311903} a^{21} + \frac{39086419}{1836311903} a^{19} - \frac{351786396}{1836311903} a^{17} - \frac{676438256}{1836311903} a^{15} + \frac{36627749}{1836311903} a^{13} - \frac{346845481}{1836311903} a^{11} - \frac{800303953}{1836311903} a^{9} + \frac{662938311}{1836311903} a^{7} - \frac{197099920}{1836311903} a^{5} + \frac{104511814}{1836311903} a^{3} - \frac{39088144}{1836311903} a$, $\frac{1}{1836311903} a^{26} + \frac{126491647}{1836311903} a^{22} + \frac{188396239}{1836311903} a^{20} + \frac{66104067}{1836311903} a^{18} - \frac{407060998}{1836311903} a^{16} + \frac{363028791}{1836311903} a^{14} + \frac{747209412}{1836311903} a^{12} - \frac{238614090}{1836311903} a^{10} + \frac{443714411}{1836311903} a^{8} + \frac{830709094}{1836311903} a^{6} - \frac{865311201}{1836311903} a^{4} - \frac{860261104}{1836311903} a^{2} + \frac{2817}{28657}$, $\frac{1}{1836311903} a^{27} - \frac{351}{1836311903} a^{23} - \frac{574911613}{1836311903} a^{21} + \frac{353934975}{1836311903} a^{19} + \frac{155877110}{1836311903} a^{17} - \frac{8681156}{1836311903} a^{15} - \frac{833883249}{1836311903} a^{13} + \frac{883307460}{1836311903} a^{11} + \frac{63235422}{1836311903} a^{9} - \frac{410053035}{1836311903} a^{7} + \frac{160380143}{1836311903} a^{5} - \frac{594295487}{1836311903} a^{3} - \frac{582797309}{1836311903} a$, $\frac{1}{1836311903} a^{28} - \frac{703590318}{1836311903} a^{22} - \frac{487854197}{1836311903} a^{20} + \frac{807933406}{1836311903} a^{18} - \frac{707225497}{1836311903} a^{16} + \frac{884140812}{1836311903} a^{14} - \frac{723683826}{1836311903} a^{12} + \frac{897923391}{1836311903} a^{10} + \frac{184667215}{1836311903} a^{8} + \frac{781753189}{1836311903} a^{6} + \frac{847146987}{1836311903} a^{4} + \frac{374855273}{1836311903} a^{2} + \frac{4016}{28657}$, $\frac{1}{1836311903} a^{29} - \frac{3654}{1836311903} a^{23} - \frac{143540342}{1836311903} a^{21} - \frac{798893241}{1836311903} a^{19} + \frac{555545208}{1836311903} a^{17} + \frac{423966706}{1836311903} a^{15} - \frac{262048213}{1836311903} a^{13} - \frac{24144241}{1836311903} a^{11} + \frac{303278538}{1836311903} a^{9} - \frac{23938366}{1836311903} a^{7} + \frac{503606871}{1836311903} a^{5} + \frac{145198075}{1836311903} a^{3} + \frac{601655119}{1836311903} a$, $\frac{1}{1836311903} a^{30} + \frac{400278168}{1836311903} a^{22} + \frac{90274356}{1836311903} a^{20} + \frac{375050196}{1836311903} a^{18} - \frac{162000274}{1836311903} a^{16} - \frac{740096249}{1836311903} a^{14} + \frac{197235835}{1836311903} a^{12} - \frac{895239006}{1836311903} a^{10} - \frac{848401239}{1836311903} a^{8} + \frac{3691359}{1836311903} a^{6} - \frac{575379647}{1836311903} a^{4} + \frac{306551361}{1836311903} a^{2} + \frac{11681}{28657}$, $\frac{1}{1836311903} a^{31} - \frac{2808}{1836311903} a^{23} + \frac{115177289}{1836311903} a^{21} + \frac{126020866}{1836311903} a^{19} - \frac{578844996}{1836311903} a^{17} - \frac{311358872}{1836311903} a^{15} - \frac{803151378}{1836311903} a^{13} - \frac{661455200}{1836311903} a^{11} + \frac{430388729}{1836311903} a^{9} + \frac{338204520}{1836311903} a^{7} - \frac{686884034}{1836311903} a^{5} - \frac{241313165}{1836311903} a^{3} + \frac{773409732}{1836311903} a$, $\frac{1}{1836311903} a^{32} - \frac{914252351}{1836311903} a^{22} + \frac{736955102}{1836311903} a^{20} - \frac{871330337}{1836311903} a^{18} - \frac{390777891}{1836311903} a^{16} + \frac{86857789}{1836311903} a^{14} - \frac{663202167}{1836311903} a^{12} - \frac{237355131}{1836311903} a^{10} - \frac{412969189}{1836311903} a^{8} + \frac{611476528}{1836311903} a^{6} + \frac{272355209}{1836311903} a^{4} - \frac{746929127}{1836311903} a^{2} + \frac{3471}{28657}$, $\frac{1}{1836311903} a^{33} - \frac{8080}{1836311903} a^{23} - \frac{91232198}{1836311903} a^{21} + \frac{65295051}{1836311903} a^{19} + \frac{146655487}{1836311903} a^{17} + \frac{96371913}{1836311903} a^{15} - \frac{73297822}{1836311903} a^{13} + \frac{372695085}{1836311903} a^{11} + \frac{561855980}{1836311903} a^{9} - \frac{340680954}{1836311903} a^{7} - \frac{634466899}{1836311903} a^{5} - \frac{889927557}{1836311903} a^{3} - \frac{605769091}{1836311903} a$, $\frac{1}{1836311903} a^{34} - \frac{615456100}{1836311903} a^{22} + \frac{578119397}{1836311903} a^{20} + \frac{790821829}{1836311903} a^{18} + \frac{485179877}{1836311903} a^{16} - \frac{719309041}{1836311903} a^{14} - \frac{804223157}{1836311903} a^{12} - \frac{62121076}{1836311903} a^{10} + \frac{422318783}{1836311903} a^{8} - \frac{84523256}{1836311903} a^{6} - \frac{2355328}{13210877} a^{4} + \frac{256356172}{1836311903} a^{2} - \frac{12301}{28657}$, $\frac{1}{1836311903} a^{35} + \frac{10289}{1836311903} a^{23} - \frac{723424229}{1836311903} a^{21} - \frac{884237135}{1836311903} a^{19} - \frac{250330685}{1836311903} a^{17} + \frac{366676631}{1836311903} a^{15} + \frac{334434081}{1836311903} a^{13} - \frac{75303296}{1836311903} a^{11} + \frac{828761014}{1836311903} a^{9} + \frac{38873786}{1836311903} a^{7} + \frac{855684996}{1836311903} a^{5} - \frac{490674504}{1836311903} a^{3} - \frac{253467502}{1836311903} a$, $\frac{1}{1836311903} a^{36} - \frac{634502246}{1836311903} a^{22} + \frac{834942809}{1836311903} a^{20} - \frac{64725612}{1836311903} a^{18} - \frac{25703587}{1836311903} a^{16} - \frac{242219750}{1836311903} a^{14} + \frac{212953461}{1836311903} a^{12} + \frac{594265919}{1836311903} a^{10} + \frac{283378823}{1836311903} a^{8} + \frac{716512217}{1836311903} a^{6} + \frac{537264979}{1836311903} a^{4} - \frac{540176617}{1836311903} a^{2} - \frac{2768}{28657}$, $\frac{1}{1836311903} a^{37} - \frac{7609}{1836311903} a^{23} - \frac{904250521}{1836311903} a^{21} + \frac{800400561}{1836311903} a^{19} + \frac{1119365}{1836311903} a^{17} + \frac{145022291}{1836311903} a^{15} + \frac{533596634}{1836311903} a^{13} - \frac{328202911}{1836311903} a^{11} - \frac{56693796}{1836311903} a^{9} - \frac{548493734}{1836311903} a^{7} - \frac{265274306}{1836311903} a^{5} - \frac{840473320}{1836311903} a^{3} - \frac{80252099}{1836311903} a$, $\frac{1}{1836311903} a^{38} + \frac{235219487}{1836311903} a^{22} - \frac{397984960}{1836311903} a^{20} - \frac{543595289}{1836311903} a^{18} - \frac{427898623}{1836311903} a^{16} + \frac{642722539}{1836311903} a^{14} - \frac{7920025}{1836311903} a^{12} - \frac{906108738}{1836311903} a^{10} + \frac{672060473}{1836311903} a^{8} - \frac{863038409}{1836311903} a^{6} - \frac{736852163}{1836311903} a^{4} + \frac{93226984}{1836311903} a^{2} - \frac{6913}{28657}$, $\frac{1}{1836311903} a^{39} + \frac{2831}{1836311903} a^{23} - \frac{496937581}{1836311903} a^{21} + \frac{445930921}{1836311903} a^{19} - \frac{559262311}{1836311903} a^{17} + \frac{629626290}{1836311903} a^{15} + \frac{634741857}{1836311903} a^{13} - \frac{484764867}{1836311903} a^{11} - \frac{299211228}{1836311903} a^{9} - \frac{647537769}{1836311903} a^{7} - \frac{196581742}{1836311903} a^{5} + \frac{433872743}{1836311903} a^{3} - \frac{541930748}{1836311903} a$, $\frac{1}{1836311903} a^{40} + \frac{138086343}{1836311903} a^{22} - \frac{669630730}{1836311903} a^{20} - \frac{63616387}{1836311903} a^{18} + \frac{681575660}{1836311903} a^{16} - \frac{775913697}{1836311903} a^{14} - \frac{534666212}{1836311903} a^{12} + \frac{439397767}{1836311903} a^{10} + \frac{305321928}{1836311903} a^{8} - \frac{677667170}{1836311903} a^{6} - \frac{289345599}{1836311903} a^{4} - \frac{836949430}{1836311903} a^{2} + \frac{8839}{28657}$, $\frac{1}{1836311903} a^{41} - \frac{11740}{1836311903} a^{23} + \frac{670313276}{1836311903} a^{21} - \frac{1320766}{3983323} a^{19} - \frac{66715924}{1836311903} a^{17} - \frac{514017374}{1836311903} a^{15} + \frac{78450303}{1836311903} a^{13} + \frac{41781892}{1836311903} a^{11} - \frac{189527148}{1836311903} a^{9} - \frac{748020105}{1836311903} a^{7} + \frac{346209347}{1836311903} a^{5} + \frac{901584789}{1836311903} a^{3} + \frac{70026384}{1836311903} a$, $\frac{1}{1836311903} a^{42} + \frac{154079148}{1836311903} a^{22} - \frac{272833968}{1836311903} a^{20} + \frac{423796641}{1836311903} a^{18} + \frac{14546437}{1836311903} a^{16} + \frac{207966916}{1836311903} a^{14} + \frac{377151566}{1836311903} a^{12} - \frac{114356754}{1836311903} a^{10} - \frac{862096977}{1836311903} a^{8} + \frac{422554213}{1836311903} a^{6} + \frac{878048889}{1836311903} a^{4} + \frac{72703998}{1836311903} a^{2} - \frac{12553}{28657}$, $\frac{1}{1836311903} a^{43} - \frac{9541}{1836311903} a^{23} - \frac{401417927}{1836311903} a^{21} - \frac{126675672}{1836311903} a^{19} + \frac{30508386}{1836311903} a^{17} + \frac{730727910}{1836311903} a^{15} - \frac{230520119}{1836311903} a^{13} - \frac{157141655}{1836311903} a^{11} + \frac{852279391}{1836311903} a^{9} + \frac{495715534}{1836311903} a^{7} + \frac{853661782}{1836311903} a^{5} - \frac{771072710}{1836311903} a^{3} + \frac{903344257}{1836311903} a$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{1}{64079} a^{23} + \frac{23}{64079} a^{21} - \frac{230}{64079} a^{19} + \frac{1311}{64079} a^{17} - \frac{4692}{64079} a^{15} + \frac{10948}{64079} a^{13} - \frac{16744}{64079} a^{11} + \frac{16445}{64079} a^{9} - \frac{9867}{64079} a^{7} + \frac{3289}{64079} a^{5} - \frac{506}{64079} a^{3} + \frac{23}{64079} a \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_2\times C_{22}$ (as 44T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 44
The 44 conjugacy class representatives for $C_2\times C_{22}$
Character table for $C_2\times C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{115}) \), \(\Q(\sqrt{-115}) \), \(\Q(i, \sqrt{115})\), \(\Q(\zeta_{23})^+\), 22.0.7198079267989980836471065337135104.1, 22.22.8083780427918435509708715954790400000000000.1, 22.0.1927323443393334271838358868310546875.1

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 R $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{4}$ $22^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{4}$ $22^{2}$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed
23Data not computed