Properties

Label 44.0.860...376.2
Degree $44$
Signature $[0, 22]$
Discriminant $8.601\times 10^{80}$
Root discriminant $69.09$
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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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 23*x^42 + 299*x^40 + 2668*x^38 + 18055*x^36 + 96646*x^34 + 421245*x^32 + 1516689*x^30 + 4557519*x^28 + 11467961*x^26 + 24199128*x^24 + 42662286*x^22 + 62532561*x^20 + 75392022*x^18 + 73935156*x^16 + 57768387*x^14 + 35301228*x^12 + 16195335*x^10 + 5446584*x^8 + 1210352*x^6 + 180389*x^4 + 11638*x^2 + 529)
 
gp: K = bnfinit(x^44 + 23*x^42 + 299*x^40 + 2668*x^38 + 18055*x^36 + 96646*x^34 + 421245*x^32 + 1516689*x^30 + 4557519*x^28 + 11467961*x^26 + 24199128*x^24 + 42662286*x^22 + 62532561*x^20 + 75392022*x^18 + 73935156*x^16 + 57768387*x^14 + 35301228*x^12 + 16195335*x^10 + 5446584*x^8 + 1210352*x^6 + 180389*x^4 + 11638*x^2 + 529, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, 0, 11638, 0, 180389, 0, 1210352, 0, 5446584, 0, 16195335, 0, 35301228, 0, 57768387, 0, 73935156, 0, 75392022, 0, 62532561, 0, 42662286, 0, 24199128, 0, 11467961, 0, 4557519, 0, 1516689, 0, 421245, 0, 96646, 0, 18055, 0, 2668, 0, 299, 0, 23, 0, 1]);
 

\( x^{44} + 23 x^{42} + 299 x^{40} + 2668 x^{38} + 18055 x^{36} + 96646 x^{34} + 421245 x^{32} + 1516689 x^{30} + 4557519 x^{28} + 11467961 x^{26} + 24199128 x^{24} + 42662286 x^{22} + 62532561 x^{20} + 75392022 x^{18} + 73935156 x^{16} + 57768387 x^{14} + 35301228 x^{12} + 16195335 x^{10} + 5446584 x^{8} + 1210352 x^{6} + 180389 x^{4} + 11638 x^{2} + 529 \)

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:  \(860\!\cdots\!376\)\(\medspace = 2^{44}\cdot 3^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $69.09$
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:  $2, 3, 23$
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:  \(276=2^{2}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{276}(1,·)$, $\chi_{276}(107,·)$, $\chi_{276}(133,·)$, $\chi_{276}(7,·)$, $\chi_{276}(265,·)$, $\chi_{276}(11,·)$, $\chi_{276}(13,·)$, $\chi_{276}(143,·)$, $\chi_{276}(19,·)$, $\chi_{276}(269,·)$, $\chi_{276}(25,·)$, $\chi_{276}(155,·)$, $\chi_{276}(29,·)$, $\chi_{276}(41,·)$, $\chi_{276}(43,·)$, $\chi_{276}(257,·)$, $\chi_{276}(173,·)$, $\chi_{276}(175,·)$, $\chi_{276}(49,·)$, $\chi_{276}(185,·)$, $\chi_{276}(191,·)$, $\chi_{276}(193,·)$, $\chi_{276}(67,·)$, $\chi_{276}(197,·)$, $\chi_{276}(199,·)$, $\chi_{276}(73,·)$, $\chi_{276}(203,·)$, $\chi_{276}(77,·)$, $\chi_{276}(79,·)$, $\chi_{276}(209,·)$, $\chi_{276}(83,·)$, $\chi_{276}(85,·)$, $\chi_{276}(91,·)$, $\chi_{276}(263,·)$, $\chi_{276}(227,·)$, $\chi_{276}(101,·)$, $\chi_{276}(103,·)$, $\chi_{276}(233,·)$, $\chi_{276}(235,·)$, $\chi_{276}(275,·)$, $\chi_{276}(169,·)$, $\chi_{276}(121,·)$, $\chi_{276}(251,·)$, $\chi_{276}(247,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23} a^{22}$, $\frac{1}{23} a^{23}$, $\frac{1}{23} a^{24}$, $\frac{1}{23} a^{25}$, $\frac{1}{23} a^{26}$, $\frac{1}{23} a^{27}$, $\frac{1}{23} a^{28}$, $\frac{1}{23} a^{29}$, $\frac{1}{23} a^{30}$, $\frac{1}{23} a^{31}$, $\frac{1}{23} a^{32}$, $\frac{1}{23} a^{33}$, $\frac{1}{23} a^{34}$, $\frac{1}{23} a^{35}$, $\frac{1}{23} a^{36}$, $\frac{1}{23} a^{37}$, $\frac{1}{23} a^{38}$, $\frac{1}{23} a^{39}$, $\frac{1}{24556157} a^{40} + \frac{305889}{24556157} a^{38} - \frac{203316}{24556157} a^{36} - \frac{15232}{24556157} a^{34} + \frac{423632}{24556157} a^{32} - \frac{263735}{24556157} a^{30} - \frac{17058}{1067659} a^{28} + \frac{79693}{24556157} a^{26} + \frac{384404}{24556157} a^{24} + \frac{210138}{24556157} a^{22} + \frac{392320}{1067659} a^{20} - \frac{491477}{1067659} a^{18} + \frac{460291}{1067659} a^{16} + \frac{409417}{1067659} a^{14} - \frac{360327}{1067659} a^{12} + \frac{372646}{1067659} a^{10} - \frac{195531}{1067659} a^{8} + \frac{255097}{1067659} a^{6} + \frac{12929}{1067659} a^{4} + \frac{446170}{1067659} a^{2} + \frac{55697}{1067659}$, $\frac{1}{24556157} a^{41} + \frac{305889}{24556157} a^{39} - \frac{203316}{24556157} a^{37} - \frac{15232}{24556157} a^{35} + \frac{423632}{24556157} a^{33} - \frac{263735}{24556157} a^{31} - \frac{17058}{1067659} a^{29} + \frac{79693}{24556157} a^{27} + \frac{384404}{24556157} a^{25} + \frac{210138}{24556157} a^{23} + \frac{392320}{1067659} a^{21} - \frac{491477}{1067659} a^{19} + \frac{460291}{1067659} a^{17} + \frac{409417}{1067659} a^{15} - \frac{360327}{1067659} a^{13} + \frac{372646}{1067659} a^{11} - \frac{195531}{1067659} a^{9} + \frac{255097}{1067659} a^{7} + \frac{12929}{1067659} a^{5} + \frac{446170}{1067659} a^{3} + \frac{55697}{1067659} a$, $\frac{1}{636968065704100522440858389810697591023} a^{42} + \frac{5755636131279205685327418474843}{636968065704100522440858389810697591023} a^{40} - \frac{2766808978853660638281687333283189226}{636968065704100522440858389810697591023} a^{38} + \frac{8420125655964967570434335740564366516}{636968065704100522440858389810697591023} a^{36} + \frac{5363834181750727225775342499545938237}{636968065704100522440858389810697591023} a^{34} - \frac{5011363664417770982567818120174250979}{636968065704100522440858389810697591023} a^{32} + \frac{6462774396101919123854667907489977390}{636968065704100522440858389810697591023} a^{30} + \frac{1361369016049795655998233811816501799}{636968065704100522440858389810697591023} a^{28} + \frac{13749647834678826284671003425575098586}{636968065704100522440858389810697591023} a^{26} + \frac{9151572527542995230221480787613417001}{636968065704100522440858389810697591023} a^{24} + \frac{11197155162359815775484628671272258602}{636968065704100522440858389810697591023} a^{22} - \frac{13338061847948670589198734037381162771}{27694263726265240106124277817856417001} a^{20} + \frac{6650709677926776184583509957164195314}{27694263726265240106124277817856417001} a^{18} - \frac{6820424214318991627456015401316240546}{27694263726265240106124277817856417001} a^{16} + \frac{10544871495508316610122098060856487900}{27694263726265240106124277817856417001} a^{14} - \frac{1169895739344639205339676515850645904}{27694263726265240106124277817856417001} a^{12} - \frac{91687202865078033400874710337434905}{27694263726265240106124277817856417001} a^{10} - \frac{1264766535778439148738828954779831460}{27694263726265240106124277817856417001} a^{8} + \frac{406874121304761360307825622872948776}{27694263726265240106124277817856417001} a^{6} - \frac{3615734122086171837070622535576165639}{27694263726265240106124277817856417001} a^{4} - \frac{1644145403094707030709335232608519251}{27694263726265240106124277817856417001} a^{2} - \frac{2178959152626519883114055851297237383}{27694263726265240106124277817856417001}$, $\frac{1}{636968065704100522440858389810697591023} a^{43} + \frac{5755636131279205685327418474843}{636968065704100522440858389810697591023} a^{41} - \frac{2766808978853660638281687333283189226}{636968065704100522440858389810697591023} a^{39} + \frac{8420125655964967570434335740564366516}{636968065704100522440858389810697591023} a^{37} + \frac{5363834181750727225775342499545938237}{636968065704100522440858389810697591023} a^{35} - \frac{5011363664417770982567818120174250979}{636968065704100522440858389810697591023} a^{33} + \frac{6462774396101919123854667907489977390}{636968065704100522440858389810697591023} a^{31} + \frac{1361369016049795655998233811816501799}{636968065704100522440858389810697591023} a^{29} + \frac{13749647834678826284671003425575098586}{636968065704100522440858389810697591023} a^{27} + \frac{9151572527542995230221480787613417001}{636968065704100522440858389810697591023} a^{25} + \frac{11197155162359815775484628671272258602}{636968065704100522440858389810697591023} a^{23} - \frac{13338061847948670589198734037381162771}{27694263726265240106124277817856417001} a^{21} + \frac{6650709677926776184583509957164195314}{27694263726265240106124277817856417001} a^{19} - \frac{6820424214318991627456015401316240546}{27694263726265240106124277817856417001} a^{17} + \frac{10544871495508316610122098060856487900}{27694263726265240106124277817856417001} a^{15} - \frac{1169895739344639205339676515850645904}{27694263726265240106124277817856417001} a^{13} - \frac{91687202865078033400874710337434905}{27694263726265240106124277817856417001} a^{11} - \frac{1264766535778439148738828954779831460}{27694263726265240106124277817856417001} a^{9} + \frac{406874121304761360307825622872948776}{27694263726265240106124277817856417001} a^{7} - \frac{3615734122086171837070622535576165639}{27694263726265240106124277817856417001} a^{5} - \frac{1644145403094707030709335232608519251}{27694263726265240106124277817856417001} a^{3} - \frac{2178959152626519883114055851297237383}{27694263726265240106124277817856417001} 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{122886834586606027611739775648793658}{636968065704100522440858389810697591023} a^{42} - \frac{2811939192352668411885229675135639801}{636968065704100522440858389810697591023} a^{40} - \frac{36414697296856284759939776538503707259}{636968065704100522440858389810697591023} a^{38} - \frac{323631039610506013057573865960345462816}{636968065704100522440858389810697591023} a^{36} - \frac{94840309042035377828763732789936159253}{27694263726265240106124277817856417001} a^{34} - \frac{11625862554251792319858035429465402898456}{636968065704100522440858389810697591023} a^{32} - \frac{50437469985103885077225421117261836876234}{636968065704100522440858389810697591023} a^{30} - \frac{180656467729518231288054018432112012295220}{636968065704100522440858389810697591023} a^{28} - \frac{539700580015441526193324798291357128560367}{636968065704100522440858389810697591023} a^{26} - \frac{1348925463646387817733501972052698954948762}{636968065704100522440858389810697591023} a^{24} - \frac{2824335798154840894937367872769089973737085}{636968065704100522440858389810697591023} a^{22} - \frac{214483121678187214802353819142117544539397}{27694263726265240106124277817856417001} a^{20} - \frac{310899250664862701002677708578524574741901}{27694263726265240106124277817856417001} a^{18} - \frac{369687660510746256879893799985492355417343}{27694263726265240106124277817856417001} a^{16} - \frac{356393319410208792238368353696847232360374}{27694263726265240106124277817856417001} a^{14} - \frac{272294467684043844839338109120170952735511}{27694263726265240106124277817856417001} a^{12} - \frac{161725619685950643130152670201485133448169}{27694263726265240106124277817856417001} a^{10} - \frac{71231131240740897059214689752582901822260}{27694263726265240106124277817856417001} a^{8} - \frac{22806403126614556303212626774792213774456}{27694263726265240106124277817856417001} a^{6} - \frac{4627967503254677839370131558107172115801}{27694263726265240106124277817856417001} a^{4} - \frac{692427001424194557692742582610301257789}{27694263726265240106124277817856417001} a^{2} - \frac{16470264746805560920516187670108347434}{27694263726265240106124277817856417001} \) (order $6$)
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

\(\Q(\sqrt{-69}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{23})\), \(\Q(\zeta_{23})^+\), 22.0.29327717405992286110481815659381862170624.1, \(\Q(\zeta_{92})^+\), 22.0.304011857053427966889939263171547.1

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 $22^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{4}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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

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
3Data not computed
23Data not computed