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Properties

Label 44.0.162...944.1
Degree $44$
Signature $[0, 22]$
Discriminant $1.626\times 10^{78}$
Root discriminant $59.91$
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 - 21*x^42 + 251*x^40 - 2052*x^38 + 12691*x^36 - 61778*x^34 + 243629*x^32 - 788303*x^30 + 2113175*x^28 - 4700059*x^26 + 8677408*x^24 - 13214290*x^22 + 16492213*x^20 - 16617826*x^18 + 13339732*x^16 - 8284333*x^14 + 3900832*x^12 - 1305733*x^10 + 306592*x^8 - 41184*x^6 + 3641*x^4 - 66*x^2 + 1)
 
gp: K = bnfinit(x^44 - 21*x^42 + 251*x^40 - 2052*x^38 + 12691*x^36 - 61778*x^34 + 243629*x^32 - 788303*x^30 + 2113175*x^28 - 4700059*x^26 + 8677408*x^24 - 13214290*x^22 + 16492213*x^20 - 16617826*x^18 + 13339732*x^16 - 8284333*x^14 + 3900832*x^12 - 1305733*x^10 + 306592*x^8 - 41184*x^6 + 3641*x^4 - 66*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -66, 0, 3641, 0, -41184, 0, 306592, 0, -1305733, 0, 3900832, 0, -8284333, 0, 13339732, 0, -16617826, 0, 16492213, 0, -13214290, 0, 8677408, 0, -4700059, 0, 2113175, 0, -788303, 0, 243629, 0, -61778, 0, 12691, 0, -2052, 0, 251, 0, -21, 0, 1]);
 

\( x^{44} - 21 x^{42} + 251 x^{40} - 2052 x^{38} + 12691 x^{36} - 61778 x^{34} + 243629 x^{32} - 788303 x^{30} + 2113175 x^{28} - 4700059 x^{26} + 8677408 x^{24} - 13214290 x^{22} + 16492213 x^{20} - 16617826 x^{18} + 13339732 x^{16} - 8284333 x^{14} + 3900832 x^{12} - 1305733 x^{10} + 306592 x^{8} - 41184 x^{6} + 3641 x^{4} - 66 x^{2} + 1 \)

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:  \(162\!\cdots\!944\)\(\medspace = 2^{44}\cdot 3^{22}\cdot 23^{40}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $59.91$
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$$\chi_{276}(1,·)$, $\chi_{276}(131,·)$, $\chi_{276}(133,·)$, $\chi_{276}(257,·)$, $\chi_{276}(265,·)$, $\chi_{276}(139,·)$, $\chi_{276}(13,·)$, $\chi_{276}(271,·)$, $\chi_{276}(259,·)$, $\chi_{276}(151,·)$, $\chi_{276}(25,·)$, $\chi_{276}(29,·)$, $\chi_{276}(31,·)$, $\chi_{276}(35,·)$, $\chi_{276}(167,·)$, $\chi_{276}(41,·)$, $\chi_{276}(95,·)$, $\chi_{276}(173,·)$, $\chi_{276}(47,·)$, $\chi_{276}(49,·)$, $\chi_{276}(179,·)$, $\chi_{276}(55,·)$, $\chi_{276}(185,·)$, $\chi_{276}(59,·)$, $\chi_{276}(193,·)$, $\chi_{276}(197,·)$, $\chi_{276}(71,·)$, $\chi_{276}(73,·)$, $\chi_{276}(119,·)$, $\chi_{276}(77,·)$, $\chi_{276}(269,·)$, $\chi_{276}(209,·)$, $\chi_{276}(163,·)$, $\chi_{276}(85,·)$, $\chi_{276}(215,·)$, $\chi_{276}(223,·)$, $\chi_{276}(187,·)$, $\chi_{276}(101,·)$, $\chi_{276}(233,·)$, $\chi_{276}(239,·)$, $\chi_{276}(211,·)$, $\chi_{276}(169,·)$, $\chi_{276}(121,·)$, $\chi_{276}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{38503} a^{40} - \frac{14974}{38503} a^{38} - \frac{7545}{38503} a^{36} - \frac{11609}{38503} a^{34} + \frac{12115}{38503} a^{32} + \frac{3952}{38503} a^{30} + \frac{7568}{38503} a^{28} - \frac{9852}{38503} a^{26} + \frac{3271}{38503} a^{24} - \frac{2590}{38503} a^{22} - \frac{4438}{38503} a^{20} + \frac{12705}{38503} a^{18} - \frac{13257}{38503} a^{16} + \frac{10861}{38503} a^{14} - \frac{18128}{38503} a^{12} - \frac{5037}{38503} a^{10} + \frac{16826}{38503} a^{8} - \frac{11491}{38503} a^{6} - \frac{15191}{38503} a^{4} - \frac{10591}{38503} a^{2} + \frac{10893}{38503}$, $\frac{1}{38503} a^{41} - \frac{14974}{38503} a^{39} - \frac{7545}{38503} a^{37} - \frac{11609}{38503} a^{35} + \frac{12115}{38503} a^{33} + \frac{3952}{38503} a^{31} + \frac{7568}{38503} a^{29} - \frac{9852}{38503} a^{27} + \frac{3271}{38503} a^{25} - \frac{2590}{38503} a^{23} - \frac{4438}{38503} a^{21} + \frac{12705}{38503} a^{19} - \frac{13257}{38503} a^{17} + \frac{10861}{38503} a^{15} - \frac{18128}{38503} a^{13} - \frac{5037}{38503} a^{11} + \frac{16826}{38503} a^{9} - \frac{11491}{38503} a^{7} - \frac{15191}{38503} a^{5} - \frac{10591}{38503} a^{3} + \frac{10893}{38503} a$, $\frac{1}{39550136075673318268146636906670404265971857953} a^{42} - \frac{190427017626724063368374735989346701359275}{39550136075673318268146636906670404265971857953} a^{40} - \frac{1572526082036525446983806027507517362195991850}{39550136075673318268146636906670404265971857953} a^{38} - \frac{6043209711280132912011332843785402439066069486}{39550136075673318268146636906670404265971857953} a^{36} + \frac{9678820460742205249663100784229475182200333298}{39550136075673318268146636906670404265971857953} a^{34} - \frac{4201109245464735669564270054376497698354553051}{39550136075673318268146636906670404265971857953} a^{32} + \frac{6509596457002940563422887376183362827233688333}{39550136075673318268146636906670404265971857953} a^{30} - \frac{14115369888379949737298339844354461890443385011}{39550136075673318268146636906670404265971857953} a^{28} + \frac{12724811426120002448333627115193665303208044206}{39550136075673318268146636906670404265971857953} a^{26} + \frac{4565183322316934206173715049595220111991996714}{39550136075673318268146636906670404265971857953} a^{24} + \frac{1011866770333132709080117347153082936581678729}{39550136075673318268146636906670404265971857953} a^{22} - \frac{12931447348997202512424621198798701576288144755}{39550136075673318268146636906670404265971857953} a^{20} + \frac{12704344750813168395565066667442999479046300236}{39550136075673318268146636906670404265971857953} a^{18} - \frac{13195457438765117428959233296641609764416909133}{39550136075673318268146636906670404265971857953} a^{16} - \frac{5677617635635486214683446015777435744449752752}{39550136075673318268146636906670404265971857953} a^{14} - \frac{6919763439996626467828935579443659919679065619}{39550136075673318268146636906670404265971857953} a^{12} - \frac{13549208359952055109743419614629027431743586784}{39550136075673318268146636906670404265971857953} a^{10} + \frac{16073345197087539024686008518179841593817771065}{39550136075673318268146636906670404265971857953} a^{8} - \frac{17173707055625732081191932776435455787730734434}{39550136075673318268146636906670404265971857953} a^{6} - \frac{14354073175149464966464365281147237239181017305}{39550136075673318268146636906670404265971857953} a^{4} - \frac{17186105840415152574291716832297879963731827238}{39550136075673318268146636906670404265971857953} a^{2} + \frac{18088117971194247705875516453344668155799592221}{39550136075673318268146636906670404265971857953}$, $\frac{1}{39550136075673318268146636906670404265971857953} a^{43} - \frac{190427017626724063368374735989346701359275}{39550136075673318268146636906670404265971857953} a^{41} - \frac{1572526082036525446983806027507517362195991850}{39550136075673318268146636906670404265971857953} a^{39} - \frac{6043209711280132912011332843785402439066069486}{39550136075673318268146636906670404265971857953} a^{37} + \frac{9678820460742205249663100784229475182200333298}{39550136075673318268146636906670404265971857953} a^{35} - \frac{4201109245464735669564270054376497698354553051}{39550136075673318268146636906670404265971857953} a^{33} + \frac{6509596457002940563422887376183362827233688333}{39550136075673318268146636906670404265971857953} a^{31} - \frac{14115369888379949737298339844354461890443385011}{39550136075673318268146636906670404265971857953} a^{29} + \frac{12724811426120002448333627115193665303208044206}{39550136075673318268146636906670404265971857953} a^{27} + \frac{4565183322316934206173715049595220111991996714}{39550136075673318268146636906670404265971857953} a^{25} + \frac{1011866770333132709080117347153082936581678729}{39550136075673318268146636906670404265971857953} a^{23} - \frac{12931447348997202512424621198798701576288144755}{39550136075673318268146636906670404265971857953} a^{21} + \frac{12704344750813168395565066667442999479046300236}{39550136075673318268146636906670404265971857953} a^{19} - \frac{13195457438765117428959233296641609764416909133}{39550136075673318268146636906670404265971857953} a^{17} - \frac{5677617635635486214683446015777435744449752752}{39550136075673318268146636906670404265971857953} a^{15} - \frac{6919763439996626467828935579443659919679065619}{39550136075673318268146636906670404265971857953} a^{13} - \frac{13549208359952055109743419614629027431743586784}{39550136075673318268146636906670404265971857953} a^{11} + \frac{16073345197087539024686008518179841593817771065}{39550136075673318268146636906670404265971857953} a^{9} - \frac{17173707055625732081191932776435455787730734434}{39550136075673318268146636906670404265971857953} a^{7} - \frac{14354073175149464966464365281147237239181017305}{39550136075673318268146636906670404265971857953} a^{5} - \frac{17186105840415152574291716832297879963731827238}{39550136075673318268146636906670404265971857953} a^{3} + \frac{18088117971194247705875516453344668155799592221}{39550136075673318268146636906670404265971857953} 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{1793299759063376835378664135048865178373688585}{39550136075673318268146636906670404265971857953} a^{43} - \frac{37524918191114127225451393946175224908107429507}{39550136075673318268146636906670404265971857953} a^{41} + \frac{447284901272128218229591301808252944778196436644}{39550136075673318268146636906670404265971857953} a^{39} - \frac{3645882836287000816414101768580673206196470843929}{39550136075673318268146636906670404265971857953} a^{37} + \frac{22480164947188205680925170226234287579609269605288}{39550136075673318268146636906670404265971857953} a^{35} - \frac{109057747165731681781192779380042573020302787841455}{39550136075673318268146636906670404265971857953} a^{33} + \frac{428454218369402889523266744705522378312522532365379}{39550136075673318268146636906670404265971857953} a^{31} - \frac{1380226261933181872822564645576170371973592506670921}{39550136075673318268146636906670404265971857953} a^{29} + \frac{3680880153832659028736431604610474723067831565699494}{39550136075673318268146636906670404265971857953} a^{27} - \frac{8135814576352391316054710021131930412486905029471824}{39550136075673318268146636906670404265971857953} a^{25} + \frac{14906096680940235386082672071811211460263861946573645}{39550136075673318268146636906670404265971857953} a^{23} - \frac{22479367063610811030950372300497610893271117186709459}{39550136075673318268146636906670404265971857953} a^{21} + \frac{27705573322205199049276348121608471713881569145400448}{39550136075673318268146636906670404265971857953} a^{19} - \frac{27444059324777363037209369190182981475233408409741898}{39550136075673318268146636906670404265971857953} a^{17} + \frac{21519004793326216917235629591870133556476659111095426}{39550136075673318268146636906670404265971857953} a^{15} - \frac{12900146249961478700999452315093622441872757730860338}{39550136075673318268146636906670404265971857953} a^{13} + \frac{5759968267918956170937644195812435949662336470191215}{39550136075673318268146636906670404265971857953} a^{11} - \frac{1750040917198848548965297061359053244112911785532794}{39550136075673318268146636906670404265971857953} a^{9} + \frac{348473274344915359648878440386430236588516863579197}{39550136075673318268146636906670404265971857953} a^{7} - \frac{26920199464397917471205835997146294435151744317003}{39550136075673318268146636906670404265971857953} a^{5} + \frac{490852689044286343341728016721638864406908608802}{39550136075673318268146636906670404265971857953} a^{3} + \frac{282675491634706018423048373549513976076133825027}{39550136075673318268146636906670404265971857953} a \) (order $12$)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), \(\Q(\zeta_{23})^+\), 22.0.7198079267989980836471065337135104.1, 22.22.1275118148086621135238339811277472268288.1, 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 $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ 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.

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