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Properties

Label 44.0.49175637595...0016.1
Degree $44$
Signature $[0, 22]$
Discriminant $2^{66}\cdot 89^{43}$
Root discriminant $227.32$
Ramified primes $2, 89$
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![373293056, 0, 28556918784, 0, 830390403072, 0, 11778002518016, 0, 86337597734912, 0, 330371237150720, 0, 739984902586368, 0, 1055898724237312, 0, 1014277563383808, 0, 680528102817792, 0, 327334436167680, 0, 115057842620416, 0, 29972558878720, 0, 5842319622656, 0, 856670243328, 0, 94579676288, 0, 7829898176, 0, 481058528, 0, 21520912, 0, 678536, 0, 14240, 0, 178, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 178*x^42 + 14240*x^40 + 678536*x^38 + 21520912*x^36 + 481058528*x^34 + 7829898176*x^32 + 94579676288*x^30 + 856670243328*x^28 + 5842319622656*x^26 + 29972558878720*x^24 + 115057842620416*x^22 + 327334436167680*x^20 + 680528102817792*x^18 + 1014277563383808*x^16 + 1055898724237312*x^14 + 739984902586368*x^12 + 330371237150720*x^10 + 86337597734912*x^8 + 11778002518016*x^6 + 830390403072*x^4 + 28556918784*x^2 + 373293056)
 
gp: K = bnfinit(x^44 + 178*x^42 + 14240*x^40 + 678536*x^38 + 21520912*x^36 + 481058528*x^34 + 7829898176*x^32 + 94579676288*x^30 + 856670243328*x^28 + 5842319622656*x^26 + 29972558878720*x^24 + 115057842620416*x^22 + 327334436167680*x^20 + 680528102817792*x^18 + 1014277563383808*x^16 + 1055898724237312*x^14 + 739984902586368*x^12 + 330371237150720*x^10 + 86337597734912*x^8 + 11778002518016*x^6 + 830390403072*x^4 + 28556918784*x^2 + 373293056, 1)
 

Normalized defining polynomial

\( x^{44} + 178 x^{42} + 14240 x^{40} + 678536 x^{38} + 21520912 x^{36} + 481058528 x^{34} + 7829898176 x^{32} + 94579676288 x^{30} + 856670243328 x^{28} + 5842319622656 x^{26} + 29972558878720 x^{24} + 115057842620416 x^{22} + 327334436167680 x^{20} + 680528102817792 x^{18} + 1014277563383808 x^{16} + 1055898724237312 x^{14} + 739984902586368 x^{12} + 330371237150720 x^{10} + 86337597734912 x^{8} + 11778002518016 x^{6} + 830390403072 x^{4} + 28556918784 x^{2} + 373293056 \)

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:  \(49175637595903736517624339652874895108812629131876151366914992073452102645525529996617219030733622870016=2^{66}\cdot 89^{43}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $227.32$
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:  $2, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(712=2^{3}\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{712}(1,·)$, $\chi_{712}(643,·)$, $\chi_{712}(265,·)$, $\chi_{712}(435,·)$, $\chi_{712}(401,·)$, $\chi_{712}(403,·)$, $\chi_{712}(601,·)$, $\chi_{712}(25,·)$, $\chi_{712}(539,·)$, $\chi_{712}(545,·)$, $\chi_{712}(411,·)$, $\chi_{712}(555,·)$, $\chi_{712}(427,·)$, $\chi_{712}(307,·)$, $\chi_{712}(99,·)$, $\chi_{712}(177,·)$, $\chi_{712}(691,·)$, $\chi_{712}(57,·)$, $\chi_{712}(187,·)$, $\chi_{712}(449,·)$, $\chi_{712}(603,·)$, $\chi_{712}(195,·)$, $\chi_{712}(673,·)$, $\chi_{712}(73,·)$, $\chi_{712}(587,·)$, $\chi_{712}(81,·)$, $\chi_{712}(339,·)$, $\chi_{712}(441,·)$, $\chi_{712}(345,·)$, $\chi_{712}(347,·)$, $\chi_{712}(97,·)$, $\chi_{712}(227,·)$, $\chi_{712}(131,·)$, $\chi_{712}(105,·)$, $\chi_{712}(107,·)$, $\chi_{712}(289,·)$, $\chi_{712}(625,·)$, $\chi_{712}(659,·)$, $\chi_{712}(707,·)$, $\chi_{712}(489,·)$, $\chi_{712}(153,·)$, $\chi_{712}(121,·)$, $\chi_{712}(217,·)$, $\chi_{712}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$, $\frac{1}{46923776} a^{36} + \frac{25}{23461888} a^{34} - \frac{29}{11730944} a^{32} + \frac{33}{5865472} a^{30} - \frac{5}{733184} a^{28} - \frac{1}{1466368} a^{26} + \frac{1}{183296} a^{24} + \frac{29}{183296} a^{22} + \frac{31}{91648} a^{20} - \frac{51}{91648} a^{18} - \frac{3}{45824} a^{16} + \frac{21}{11456} a^{14} - \frac{1}{5728} a^{12} - \frac{11}{5728} a^{10} - \frac{3}{716} a^{8} - \frac{3}{179} a^{6} + \frac{9}{358} a^{4} + \frac{77}{358} a^{2} + \frac{85}{179}$, $\frac{1}{46923776} a^{37} + \frac{25}{23461888} a^{35} - \frac{29}{11730944} a^{33} + \frac{33}{5865472} a^{31} - \frac{5}{733184} a^{29} - \frac{1}{1466368} a^{27} + \frac{1}{183296} a^{25} + \frac{29}{183296} a^{23} + \frac{31}{91648} a^{21} - \frac{51}{91648} a^{19} - \frac{3}{45824} a^{17} + \frac{21}{11456} a^{15} - \frac{1}{5728} a^{13} - \frac{11}{5728} a^{11} - \frac{3}{716} a^{9} - \frac{3}{179} a^{7} + \frac{9}{358} a^{5} + \frac{77}{358} a^{3} + \frac{85}{179} a$, $\frac{1}{93847552} a^{38} + \frac{31}{11730944} a^{34} + \frac{21}{5865472} a^{32} + \frac{25}{2932736} a^{30} - \frac{19}{1466368} a^{28} + \frac{29}{1466368} a^{26} - \frac{21}{366592} a^{24} + \frac{11}{91648} a^{22} + \frac{5}{91648} a^{20} + \frac{19}{91648} a^{18} - \frac{31}{22912} a^{16} + \frac{11}{11456} a^{14} + \frac{39}{11456} a^{12} + \frac{21}{1432} a^{10} - \frac{41}{1432} a^{8} + \frac{81}{1432} a^{6} - \frac{15}{716} a^{4} - \frac{25}{179} a^{2} + \frac{23}{179}$, $\frac{1}{93847552} a^{39} + \frac{31}{11730944} a^{35} + \frac{21}{5865472} a^{33} + \frac{25}{2932736} a^{31} - \frac{19}{1466368} a^{29} + \frac{29}{1466368} a^{27} - \frac{21}{366592} a^{25} + \frac{11}{91648} a^{23} + \frac{5}{91648} a^{21} + \frac{19}{91648} a^{19} - \frac{31}{22912} a^{17} + \frac{11}{11456} a^{15} + \frac{39}{11456} a^{13} + \frac{21}{1432} a^{11} - \frac{41}{1432} a^{9} + \frac{81}{1432} a^{7} - \frac{15}{716} a^{5} - \frac{25}{179} a^{3} + \frac{23}{179} a$, $\frac{1}{89905954816} a^{40} - \frac{25}{22476488704} a^{38} + \frac{233}{22476488704} a^{36} - \frac{22411}{11238244352} a^{34} + \frac{21273}{5619122176} a^{32} + \frac{5833}{702390272} a^{30} + \frac{14917}{702390272} a^{28} - \frac{10229}{351195136} a^{26} - \frac{2063}{43899392} a^{24} - \frac{2185}{43899392} a^{22} + \frac{15849}{87798784} a^{20} - \frac{14387}{43899392} a^{18} + \frac{21941}{21949696} a^{16} + \frac{2255}{5487424} a^{14} - \frac{19183}{2743712} a^{12} - \frac{20713}{1371856} a^{10} + \frac{12153}{1371856} a^{8} + \frac{21545}{685928} a^{6} - \frac{19671}{342964} a^{4} - \frac{17215}{85741} a^{2} - \frac{30649}{85741}$, $\frac{1}{89905954816} a^{41} - \frac{25}{22476488704} a^{39} + \frac{233}{22476488704} a^{37} - \frac{22411}{11238244352} a^{35} + \frac{21273}{5619122176} a^{33} + \frac{5833}{702390272} a^{31} + \frac{14917}{702390272} a^{29} - \frac{10229}{351195136} a^{27} - \frac{2063}{43899392} a^{25} - \frac{2185}{43899392} a^{23} + \frac{15849}{87798784} a^{21} - \frac{14387}{43899392} a^{19} + \frac{21941}{21949696} a^{17} + \frac{2255}{5487424} a^{15} - \frac{19183}{2743712} a^{13} - \frac{20713}{1371856} a^{11} + \frac{12153}{1371856} a^{9} + \frac{21545}{685928} a^{7} - \frac{19671}{342964} a^{5} - \frac{17215}{85741} a^{3} - \frac{30649}{85741} a$, $\frac{1}{3719772289900781005258017564548152361754103782844773903946406023349141504} a^{42} - \frac{3839404366284229530409537393899867218633771597432640744434587}{929943072475195251314504391137038090438525945711193475986601505837285376} a^{40} + \frac{279338721940519217687009059917462732631984039250251098936397763}{232485768118798812828626097784259522609631486427798368996650376459321344} a^{38} + \frac{4229697727873115951116815038860055973567279322103878852573601583}{464971536237597625657252195568519045219262972855596737993300752918642688} a^{36} - \frac{36596240358774686125741742712140165135603612618585910479892459389}{29060721014849851603578262223032440326203935803474796124581297057415168} a^{34} + \frac{8462902383583051320998665931845013268935327054054477488452516041}{116242884059399406414313048892129761304815743213899184498325188229660672} a^{32} - \frac{359943399760375678967407232706958673506950903630699731380547600881}{29060721014849851603578262223032440326203935803474796124581297057415168} a^{30} + \frac{616104327738972892838947796918866779810869496173014469384234217719}{29060721014849851603578262223032440326203935803474796124581297057415168} a^{28} + \frac{58234846556361893244102629227031992963201769265092114730126580573}{7265180253712462900894565555758110081550983950868699031145324264353792} a^{26} - \frac{146813435343232202896313521257143120273351545612409227531272628649}{1816295063428115725223641388939527520387745987717174757786331066088448} a^{24} - \frac{421601689221770471009540765934950513698095542762873260140376190157}{1816295063428115725223641388939527520387745987717174757786331066088448} a^{22} - \frac{117518565981027514771388974086321051784639399088944147755979284145}{454073765857028931305910347234881880096936496929293689446582766522112} a^{20} - \frac{209451347052866603393544052906702313536013039859360207726472835991}{908147531714057862611820694469763760193872993858587378893165533044224} a^{18} - \frac{206940387975130031981332741407470568361902034723249487941866875879}{227036882928514465652955173617440940048468248464646844723291383261056} a^{16} - \frac{875241438996746698457995530676084913500465935931621083853833594497}{227036882928514465652955173617440940048468248464646844723291383261056} a^{14} + \frac{60466585409600690161884940554067928522232302136838937931661601715}{113518441464257232826477586808720470024234124232323422361645691630528} a^{12} - \frac{121431506373590819154633056294475332126819039819044169513669213491}{14189805183032154103309698351090058753029265529040427795205711453816} a^{10} - \frac{10372441052805913923121478042520259327404473851178149845608347165}{1773725647879019262913712293886257344128658191130053474400713931727} a^{8} + \frac{107461234732715386627981672181150339088426202103236352439450035007}{1773725647879019262913712293886257344128658191130053474400713931727} a^{6} + \frac{396192289210241788563947865445458855817839968817628718819661083791}{7094902591516077051654849175545029376514632764520213897602855726908} a^{4} + \frac{752861951381189501111726070124087350492556649186302928822363945813}{3547451295758038525827424587772514688257316382260106948801427863454} a^{2} - \frac{82225258859283191951777519920807440004636171881109873738133652257}{1773725647879019262913712293886257344128658191130053474400713931727}$, $\frac{1}{3719772289900781005258017564548152361754103782844773903946406023349141504} a^{43} - \frac{3839404366284229530409537393899867218633771597432640744434587}{929943072475195251314504391137038090438525945711193475986601505837285376} a^{41} + \frac{279338721940519217687009059917462732631984039250251098936397763}{232485768118798812828626097784259522609631486427798368996650376459321344} a^{39} + \frac{4229697727873115951116815038860055973567279322103878852573601583}{464971536237597625657252195568519045219262972855596737993300752918642688} a^{37} - \frac{36596240358774686125741742712140165135603612618585910479892459389}{29060721014849851603578262223032440326203935803474796124581297057415168} a^{35} + \frac{8462902383583051320998665931845013268935327054054477488452516041}{116242884059399406414313048892129761304815743213899184498325188229660672} a^{33} - \frac{359943399760375678967407232706958673506950903630699731380547600881}{29060721014849851603578262223032440326203935803474796124581297057415168} a^{31} + \frac{616104327738972892838947796918866779810869496173014469384234217719}{29060721014849851603578262223032440326203935803474796124581297057415168} a^{29} + \frac{58234846556361893244102629227031992963201769265092114730126580573}{7265180253712462900894565555758110081550983950868699031145324264353792} a^{27} - \frac{146813435343232202896313521257143120273351545612409227531272628649}{1816295063428115725223641388939527520387745987717174757786331066088448} a^{25} - \frac{421601689221770471009540765934950513698095542762873260140376190157}{1816295063428115725223641388939527520387745987717174757786331066088448} a^{23} - \frac{117518565981027514771388974086321051784639399088944147755979284145}{454073765857028931305910347234881880096936496929293689446582766522112} a^{21} - \frac{209451347052866603393544052906702313536013039859360207726472835991}{908147531714057862611820694469763760193872993858587378893165533044224} a^{19} - \frac{206940387975130031981332741407470568361902034723249487941866875879}{227036882928514465652955173617440940048468248464646844723291383261056} a^{17} - \frac{875241438996746698457995530676084913500465935931621083853833594497}{227036882928514465652955173617440940048468248464646844723291383261056} a^{15} + \frac{60466585409600690161884940554067928522232302136838937931661601715}{113518441464257232826477586808720470024234124232323422361645691630528} a^{13} - \frac{121431506373590819154633056294475332126819039819044169513669213491}{14189805183032154103309698351090058753029265529040427795205711453816} a^{11} - \frac{10372441052805913923121478042520259327404473851178149845608347165}{1773725647879019262913712293886257344128658191130053474400713931727} a^{9} + \frac{107461234732715386627981672181150339088426202103236352439450035007}{1773725647879019262913712293886257344128658191130053474400713931727} a^{7} + \frac{396192289210241788563947865445458855817839968817628718819661083791}{7094902591516077051654849175545029376514632764520213897602855726908} a^{5} + \frac{752861951381189501111726070124087350492556649186302928822363945813}{3547451295758038525827424587772514688257316382260106948801427863454} a^{3} - \frac{82225258859283191951777519920807440004636171881109873738133652257}{1773725647879019262913712293886257344128658191130053474400713931727} 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:  \( -1 \) (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

\(\Q(\sqrt{89}) \), 4.0.45118016.1, 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 R $44$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{4}$ $44$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ $44$ $22^{2}$ $44$ $44$ $44$ $44$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{11}$ $44$ $44$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{4}$ $44$

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
2Data not computed
89Data not computed