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Properties

Label 44.0.36075838195...4304.3
Degree $44$
Signature $[0, 22]$
Discriminant $2^{66}\cdot 3^{22}\cdot 23^{42}$
Root discriminant $97.71$
Ramified primes $2, 3, 23$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2218786816, 0, 24406654976, 0, 189151576064, 0, 634573029376, 0, 1427789316096, 0, 2122754949120, 0, 2313501278208, 0, 1892954505216, 0, 1211353595904, 0, 617611444224, 0, 256133369856, 0, 87372361728, 0, 24779907072, 0, 5871596032, 0, 1166724864, 0, 194136192, 0, 26959680, 0, 3092672, 0, 288880, 0, 21344, 0, 1196, 0, 46, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 46*x^42 + 1196*x^40 + 21344*x^38 + 288880*x^36 + 3092672*x^34 + 26959680*x^32 + 194136192*x^30 + 1166724864*x^28 + 5871596032*x^26 + 24779907072*x^24 + 87372361728*x^22 + 256133369856*x^20 + 617611444224*x^18 + 1211353595904*x^16 + 1892954505216*x^14 + 2313501278208*x^12 + 2122754949120*x^10 + 1427789316096*x^8 + 634573029376*x^6 + 189151576064*x^4 + 24406654976*x^2 + 2218786816)
 
gp: K = bnfinit(x^44 + 46*x^42 + 1196*x^40 + 21344*x^38 + 288880*x^36 + 3092672*x^34 + 26959680*x^32 + 194136192*x^30 + 1166724864*x^28 + 5871596032*x^26 + 24779907072*x^24 + 87372361728*x^22 + 256133369856*x^20 + 617611444224*x^18 + 1211353595904*x^16 + 1892954505216*x^14 + 2313501278208*x^12 + 2122754949120*x^10 + 1427789316096*x^8 + 634573029376*x^6 + 189151576064*x^4 + 24406654976*x^2 + 2218786816, 1)
 

Normalized defining polynomial

\( x^{44} + 46 x^{42} + 1196 x^{40} + 21344 x^{38} + 288880 x^{36} + 3092672 x^{34} + 26959680 x^{32} + 194136192 x^{30} + 1166724864 x^{28} + 5871596032 x^{26} + 24779907072 x^{24} + 87372361728 x^{22} + 256133369856 x^{20} + 617611444224 x^{18} + 1211353595904 x^{16} + 1892954505216 x^{14} + 2313501278208 x^{12} + 2122754949120 x^{10} + 1427789316096 x^{8} + 634573029376 x^{6} + 189151576064 x^{4} + 24406654976 x^{2} + 2218786816 \)

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:  \(3607583819545152459027384276140645884702253651640471494457079112798455926939134201954304=2^{66}\cdot 3^{22}\cdot 23^{42}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.71$
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:  $2, 3, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(552=2^{3}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{552}(1,·)$, $\chi_{552}(107,·)$, $\chi_{552}(257,·)$, $\chi_{552}(265,·)$, $\chi_{552}(523,·)$, $\chi_{552}(19,·)$, $\chi_{552}(25,·)$, $\chi_{552}(409,·)$, $\chi_{552}(539,·)$, $\chi_{552}(545,·)$, $\chi_{552}(155,·)$, $\chi_{552}(49,·)$, $\chi_{552}(41,·)$, $\chi_{552}(43,·)$, $\chi_{552}(379,·)$, $\chi_{552}(305,·)$, $\chi_{552}(449,·)$, $\chi_{552}(283,·)$, $\chi_{552}(185,·)$, $\chi_{552}(169,·)$, $\chi_{552}(193,·)$, $\chi_{552}(83,·)$, $\chi_{552}(67,·)$, $\chi_{552}(289,·)$, $\chi_{552}(275,·)$, $\chi_{552}(73,·)$, $\chi_{552}(203,·)$, $\chi_{552}(11,·)$, $\chi_{552}(209,·)$, $\chi_{552}(419,·)$, $\chi_{552}(377,·)$, $\chi_{552}(355,·)$, $\chi_{552}(473,·)$, $\chi_{552}(475,·)$, $\chi_{552}(353,·)$, $\chi_{552}(227,·)$, $\chi_{552}(91,·)$, $\chi_{552}(233,·)$, $\chi_{552}(235,·)$, $\chi_{552}(451,·)$, $\chi_{552}(467,·)$, $\chi_{552}(361,·)$, $\chi_{552}(121,·)$, $\chi_{552}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{47104} a^{22}$, $\frac{1}{47104} a^{23}$, $\frac{1}{94208} a^{24}$, $\frac{1}{94208} a^{25}$, $\frac{1}{188416} a^{26}$, $\frac{1}{188416} a^{27}$, $\frac{1}{376832} a^{28}$, $\frac{1}{376832} a^{29}$, $\frac{1}{753664} a^{30}$, $\frac{1}{753664} a^{31}$, $\frac{1}{1507328} a^{32}$, $\frac{1}{1507328} a^{33}$, $\frac{1}{3014656} a^{34}$, $\frac{1}{3014656} a^{35}$, $\frac{1}{6029312} a^{36}$, $\frac{1}{6029312} a^{37}$, $\frac{1}{12058624} a^{38}$, $\frac{1}{12058624} a^{39}$, $\frac{1}{25748996882432} a^{40} + \frac{305889}{12874498441216} a^{38} - \frac{50829}{1609312305152} a^{36} - \frac{119}{25145504768} a^{34} + \frac{26477}{100582019072} a^{32} - \frac{263735}{804656152576} a^{30} - \frac{8529}{8746262528} a^{28} + \frac{79693}{201164038144} a^{26} + \frac{96101}{25145504768} a^{24} + \frac{105069}{25145504768} a^{22} + \frac{3065}{8541272} a^{20} - \frac{491477}{546641408} a^{18} + \frac{460291}{273320704} a^{16} + \frac{409417}{136660352} a^{14} - \frac{360327}{68330176} a^{12} + \frac{186323}{17082544} a^{10} - \frac{195531}{17082544} a^{8} + \frac{255097}{8541272} a^{6} + \frac{12929}{4270636} a^{4} + \frac{223085}{1067659} a^{2} + \frac{55697}{1067659}$, $\frac{1}{25748996882432} a^{41} + \frac{305889}{12874498441216} a^{39} - \frac{50829}{1609312305152} a^{37} - \frac{119}{25145504768} a^{35} + \frac{26477}{100582019072} a^{33} - \frac{263735}{804656152576} a^{31} - \frac{8529}{8746262528} a^{29} + \frac{79693}{201164038144} a^{27} + \frac{96101}{25145504768} a^{25} + \frac{105069}{25145504768} a^{23} + \frac{3065}{8541272} a^{21} - \frac{491477}{546641408} a^{19} + \frac{460291}{273320704} a^{17} + \frac{409417}{136660352} a^{15} - \frac{360327}{68330176} a^{13} + \frac{186323}{17082544} a^{11} - \frac{195531}{17082544} a^{9} + \frac{255097}{8541272} a^{7} + \frac{12929}{4270636} a^{5} + \frac{223085}{1067659} a^{3} + \frac{55697}{1067659} a$, $\frac{1}{1335818852927485818837891053908284074409066496} a^{42} + \frac{5755636131279205685327418474843}{667909426463742909418945526954142037204533248} a^{40} - \frac{1383404489426830319140843666641594613}{166977356615935727354736381738535509301133312} a^{38} + \frac{2105031413991241892608583935141091629}{41744339153983931838684095434633877325283328} a^{36} + \frac{5363834181750727225775342499545938237}{83488678307967863677368190869267754650566656} a^{34} - \frac{5011363664417770982567818120174250979}{41744339153983931838684095434633877325283328} a^{32} + \frac{3231387198050959561927333953744988695}{10436084788495982959671023858658469331320832} a^{30} + \frac{1361369016049795655998233811816501799}{10436084788495982959671023858658469331320832} a^{28} + \frac{6874823917339413142335501712787549293}{2609021197123995739917755964664617332830208} a^{26} + \frac{9151572527542995230221480787613417001}{2609021197123995739917755964664617332830208} a^{24} + \frac{5598577581179907887742314335636129301}{652255299280998934979438991166154333207552} a^{22} - \frac{13338061847948670589198734037381162771}{28358926055695605868671260485484971009024} a^{20} + \frac{3325354838963388092291754978582097657}{7089731513923901467167815121371242752256} a^{18} - \frac{3410212107159495813728007700658120273}{3544865756961950733583907560685621376128} a^{16} + \frac{2636217873877079152530524515214121975}{886216439240487683395976890171405344032} a^{14} - \frac{73118483709039950333729782240665369}{110777054905060960424497111271425668004} a^{12} - \frac{91687202865078033400874710337434905}{886216439240487683395976890171405344032} a^{10} - \frac{316191633944609787184707238694957865}{110777054905060960424497111271425668004} a^{8} + \frac{50859265163095170038478202859118597}{27694263726265240106124277817856417001} a^{6} - \frac{3615734122086171837070622535576165639}{110777054905060960424497111271425668004} a^{4} - \frac{1644145403094707030709335232608519251}{55388527452530480212248555635712834002} a^{2} - \frac{2178959152626519883114055851297237383}{27694263726265240106124277817856417001}$, $\frac{1}{1335818852927485818837891053908284074409066496} a^{43} + \frac{5755636131279205685327418474843}{667909426463742909418945526954142037204533248} a^{41} - \frac{1383404489426830319140843666641594613}{166977356615935727354736381738535509301133312} a^{39} + \frac{2105031413991241892608583935141091629}{41744339153983931838684095434633877325283328} a^{37} + \frac{5363834181750727225775342499545938237}{83488678307967863677368190869267754650566656} a^{35} - \frac{5011363664417770982567818120174250979}{41744339153983931838684095434633877325283328} a^{33} + \frac{3231387198050959561927333953744988695}{10436084788495982959671023858658469331320832} a^{31} + \frac{1361369016049795655998233811816501799}{10436084788495982959671023858658469331320832} a^{29} + \frac{6874823917339413142335501712787549293}{2609021197123995739917755964664617332830208} a^{27} + \frac{9151572527542995230221480787613417001}{2609021197123995739917755964664617332830208} a^{25} + \frac{5598577581179907887742314335636129301}{652255299280998934979438991166154333207552} a^{23} - \frac{13338061847948670589198734037381162771}{28358926055695605868671260485484971009024} a^{21} + \frac{3325354838963388092291754978582097657}{7089731513923901467167815121371242752256} a^{19} - \frac{3410212107159495813728007700658120273}{3544865756961950733583907560685621376128} a^{17} + \frac{2636217873877079152530524515214121975}{886216439240487683395976890171405344032} a^{15} - \frac{73118483709039950333729782240665369}{110777054905060960424497111271425668004} a^{13} - \frac{91687202865078033400874710337434905}{886216439240487683395976890171405344032} a^{11} - \frac{316191633944609787184707238694957865}{110777054905060960424497111271425668004} a^{9} + \frac{50859265163095170038478202859118597}{27694263726265240106124277817856417001} a^{7} - \frac{3615734122086171837070622535576165639}{110777054905060960424497111271425668004} a^{5} - \frac{1644145403094707030709335232608519251}{55388527452530480212248555635712834002} a^{3} - \frac{2178959152626519883114055851297237383}{27694263726265240106124277817856417001} a$

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:  \( \frac{61443417293303013805869887824396829}{667909426463742909418945526954142037204533248} a^{42} + \frac{2811939192352668411885229675135639801}{667909426463742909418945526954142037204533248} a^{40} + \frac{36414697296856284759939776538503707259}{333954713231871454709472763477071018602266624} a^{38} + \frac{10113469987828312908049183311260795713}{5218042394247991479835511929329234665660416} a^{36} + \frac{94840309042035377828763732789936159253}{3629942535129037551189921342142076289155072} a^{34} + \frac{1453232819281474039982254428683175362307}{5218042394247991479835511929329234665660416} a^{32} + \frac{25218734992551942538612710558630918438117}{10436084788495982959671023858658469331320832} a^{30} + \frac{45164116932379557822013504608028003073805}{2609021197123995739917755964664617332830208} a^{28} + \frac{539700580015441526193324798291357128560367}{5218042394247991479835511929329234665660416} a^{26} + \frac{674462731823193908866750986026349477474381}{1304510598561997869958877982332308666415104} a^{24} + \frac{2824335798154840894937367872769089973737085}{1304510598561997869958877982332308666415104} a^{22} + \frac{214483121678187214802353819142117544539397}{28358926055695605868671260485484971009024} a^{20} + \frac{310899250664862701002677708578524574741901}{14179463027847802934335630242742485504512} a^{18} + \frac{369687660510746256879893799985492355417343}{7089731513923901467167815121371242752256} a^{16} + \frac{178196659705104396119184176848423616180187}{1772432878480975366791953780342810688064} a^{14} + \frac{272294467684043844839338109120170952735511}{1772432878480975366791953780342810688064} a^{12} + \frac{161725619685950643130152670201485133448169}{886216439240487683395976890171405344032} a^{10} + \frac{17807782810185224264803672438145725455565}{110777054905060960424497111271425668004} a^{8} + \frac{2850800390826819537901578346849026721807}{27694263726265240106124277817856417001} a^{6} + \frac{4627967503254677839370131558107172115801}{110777054905060960424497111271425668004} a^{4} + \frac{692427001424194557692742582610301257789}{55388527452530480212248555635712834002} a^{2} + \frac{44164528473070801026640465487964764435}{27694263726265240106124277817856417001} \) (order $6$)
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

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{-138}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{46}) \), \(\Q(\sqrt{-3}, \sqrt{46})\), \(\Q(\zeta_{23})^+\), 22.0.60063165247472201954266758470414053725437952.1, 22.0.304011857053427966889939263171547.1, 22.22.339058325839400057321133061640411938816.1

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 R $22^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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
2Data not computed
3Data not computed
23Data not computed