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Properties

Label 44.0.41418902606...4608.1
Degree $44$
Signature $[0, 22]$
Discriminant $2^{121}\cdot 23^{42}$
Root discriminant $134.17$
Ramified primes $2, 23$
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![1083392, 0, 47669248, 0, 726956032, 0, 5732227072, 0, 27460466176, 0, 87272100864, 0, 194509897472, 0, 315651405312, 0, 382830850432, 0, 353362384896, 0, 251281669632, 0, 138711729664, 0, 59657548672, 0, 19993969920, 0, 5205551360, 0, 1046043680, 0, 160562632, 0, 18537632, 0, 1573384, 0, 94760, 0, 3818, 0, 92, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 92*x^42 + 3818*x^40 + 94760*x^38 + 1573384*x^36 + 18537632*x^34 + 160562632*x^32 + 1046043680*x^30 + 5205551360*x^28 + 19993969920*x^26 + 59657548672*x^24 + 138711729664*x^22 + 251281669632*x^20 + 353362384896*x^18 + 382830850432*x^16 + 315651405312*x^14 + 194509897472*x^12 + 87272100864*x^10 + 27460466176*x^8 + 5732227072*x^6 + 726956032*x^4 + 47669248*x^2 + 1083392)
 
gp: K = bnfinit(x^44 + 92*x^42 + 3818*x^40 + 94760*x^38 + 1573384*x^36 + 18537632*x^34 + 160562632*x^32 + 1046043680*x^30 + 5205551360*x^28 + 19993969920*x^26 + 59657548672*x^24 + 138711729664*x^22 + 251281669632*x^20 + 353362384896*x^18 + 382830850432*x^16 + 315651405312*x^14 + 194509897472*x^12 + 87272100864*x^10 + 27460466176*x^8 + 5732227072*x^6 + 726956032*x^4 + 47669248*x^2 + 1083392, 1)
 

Normalized defining polynomial

\( x^{44} + 92 x^{42} + 3818 x^{40} + 94760 x^{38} + 1573384 x^{36} + 18537632 x^{34} + 160562632 x^{32} + 1046043680 x^{30} + 5205551360 x^{28} + 19993969920 x^{26} + 59657548672 x^{24} + 138711729664 x^{22} + 251281669632 x^{20} + 353362384896 x^{18} + 382830850432 x^{16} + 315651405312 x^{14} + 194509897472 x^{12} + 87272100864 x^{10} + 27460466176 x^{8} + 5732227072 x^{6} + 726956032 x^{4} + 47669248 x^{2} + 1083392 \)

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:  \(4141890260646712580912980965306954513336276372715662057543551492310346739946349214617837764608=2^{121}\cdot 23^{42}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.17$
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:  $2, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(368=2^{4}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{368}(1,·)$, $\chi_{368}(5,·)$, $\chi_{368}(257,·)$, $\chi_{368}(9,·)$, $\chi_{368}(109,·)$, $\chi_{368}(237,·)$, $\chi_{368}(21,·)$, $\chi_{368}(25,·)$, $\chi_{368}(157,·)$, $\chi_{368}(289,·)$, $\chi_{368}(37,·)$, $\chi_{368}(305,·)$, $\chi_{368}(41,·)$, $\chi_{368}(45,·)$, $\chi_{368}(49,·)$, $\chi_{368}(309,·)$, $\chi_{368}(265,·)$, $\chi_{368}(185,·)$, $\chi_{368}(61,·)$, $\chi_{368}(53,·)$, $\chi_{368}(193,·)$, $\chi_{368}(181,·)$, $\chi_{368}(353,·)$, $\chi_{368}(73,·)$, $\chi_{368}(205,·)$, $\chi_{368}(333,·)$, $\chi_{368}(81,·)$, $\chi_{368}(341,·)$, $\chi_{368}(105,·)$, $\chi_{368}(361,·)$, $\chi_{368}(221,·)$, $\chi_{368}(293,·)$, $\chi_{368}(225,·)$, $\chi_{368}(229,·)$, $\chi_{368}(209,·)$, $\chi_{368}(233,·)$, $\chi_{368}(365,·)$, $\chi_{368}(177,·)$, $\chi_{368}(189,·)$, $\chi_{368}(245,·)$, $\chi_{368}(169,·)$, $\chi_{368}(121,·)$, $\chi_{368}(125,·)$, $\chi_{368}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{16} a^{18}$, $\frac{1}{16} a^{19}$, $\frac{1}{32} a^{20}$, $\frac{1}{32} a^{21}$, $\frac{1}{736} a^{22}$, $\frac{1}{736} a^{23}$, $\frac{1}{1472} a^{24}$, $\frac{1}{1472} a^{25}$, $\frac{1}{1472} a^{26}$, $\frac{1}{1472} a^{27}$, $\frac{1}{2944} a^{28}$, $\frac{1}{2944} a^{29}$, $\frac{1}{2944} a^{30}$, $\frac{1}{2944} a^{31}$, $\frac{1}{5888} a^{32}$, $\frac{1}{5888} a^{33}$, $\frac{1}{276736} a^{34} - \frac{19}{276736} a^{32} - \frac{1}{8648} a^{30} - \frac{3}{69184} a^{28} - \frac{11}{69184} a^{26} - \frac{11}{34592} a^{24} - \frac{1}{4324} a^{22} - \frac{1}{188} a^{20} - \frac{1}{47} a^{18} - \frac{11}{376} a^{16} + \frac{3}{188} a^{14} - \frac{1}{47} a^{12} + \frac{3}{47} a^{10} + \frac{1}{47} a^{8} - \frac{21}{94} a^{6} - \frac{3}{47} a^{4} - \frac{10}{47} a^{2} + \frac{2}{47}$, $\frac{1}{276736} a^{35} - \frac{19}{276736} a^{33} - \frac{1}{8648} a^{31} - \frac{3}{69184} a^{29} - \frac{11}{69184} a^{27} - \frac{11}{34592} a^{25} - \frac{1}{4324} a^{23} - \frac{1}{188} a^{21} - \frac{1}{47} a^{19} - \frac{11}{376} a^{17} + \frac{3}{188} a^{15} - \frac{1}{47} a^{13} + \frac{3}{47} a^{11} + \frac{1}{47} a^{9} - \frac{21}{94} a^{7} - \frac{3}{47} a^{5} - \frac{10}{47} a^{3} + \frac{2}{47} a$, $\frac{1}{553472} a^{36} + \frac{15}{276736} a^{32} - \frac{7}{69184} a^{30} - \frac{21}{138368} a^{28} + \frac{1}{34592} a^{26} + \frac{9}{34592} a^{24} + \frac{5}{8648} a^{22} + \frac{1}{752} a^{20} - \frac{11}{376} a^{18} - \frac{15}{752} a^{16} + \frac{3}{188} a^{14} - \frac{17}{376} a^{12} + \frac{11}{94} a^{10} + \frac{17}{188} a^{8} + \frac{9}{94} a^{6} - \frac{10}{47} a^{4} + \frac{19}{47}$, $\frac{1}{553472} a^{37} + \frac{15}{276736} a^{33} - \frac{7}{69184} a^{31} - \frac{21}{138368} a^{29} + \frac{1}{34592} a^{27} + \frac{9}{34592} a^{25} + \frac{5}{8648} a^{23} + \frac{1}{752} a^{21} - \frac{11}{376} a^{19} - \frac{15}{752} a^{17} + \frac{3}{188} a^{15} - \frac{17}{376} a^{13} + \frac{11}{94} a^{11} + \frac{17}{188} a^{9} + \frac{9}{94} a^{7} - \frac{10}{47} a^{5} + \frac{19}{47} a$, $\frac{1}{553472} a^{38} + \frac{11}{138368} a^{32} - \frac{1}{8648} a^{30} - \frac{5}{69184} a^{26} - \frac{3}{34592} a^{24} - \frac{11}{17296} a^{22} - \frac{9}{752} a^{20} - \frac{5}{376} a^{18} + \frac{13}{752} a^{16} - \frac{13}{376} a^{14} + \frac{23}{376} a^{12} - \frac{11}{94} a^{10} + \frac{5}{188} a^{8} + \frac{13}{94} a^{6} - \frac{2}{47} a^{4} - \frac{19}{47} a^{2} + \frac{17}{47}$, $\frac{1}{553472} a^{39} + \frac{11}{138368} a^{33} - \frac{1}{8648} a^{31} - \frac{5}{69184} a^{27} - \frac{3}{34592} a^{25} - \frac{11}{17296} a^{23} - \frac{9}{752} a^{21} - \frac{5}{376} a^{19} + \frac{13}{752} a^{17} - \frac{13}{376} a^{15} + \frac{23}{376} a^{13} - \frac{11}{94} a^{11} + \frac{5}{188} a^{9} + \frac{13}{94} a^{7} - \frac{2}{47} a^{5} - \frac{19}{47} a^{3} + \frac{17}{47} a$, $\frac{1}{1106944} a^{40} + \frac{5}{276736} a^{32} - \frac{3}{34592} a^{30} + \frac{7}{69184} a^{28} - \frac{1}{3008} a^{26} - \frac{15}{69184} a^{24} + \frac{11}{17296} a^{22} - \frac{1}{94} a^{20} + \frac{9}{376} a^{18} - \frac{3}{376} a^{16} + \frac{2}{47} a^{14} + \frac{19}{376} a^{12} - \frac{3}{47} a^{10} + \frac{4}{47} a^{8} - \frac{3}{47} a^{6} + \frac{1}{47} a^{2} - \frac{22}{47}$, $\frac{1}{1106944} a^{41} + \frac{5}{276736} a^{33} - \frac{3}{34592} a^{31} + \frac{7}{69184} a^{29} - \frac{1}{3008} a^{27} - \frac{15}{69184} a^{25} + \frac{11}{17296} a^{23} - \frac{1}{94} a^{21} + \frac{9}{376} a^{19} - \frac{3}{376} a^{17} + \frac{2}{47} a^{15} + \frac{19}{376} a^{13} - \frac{3}{47} a^{11} + \frac{4}{47} a^{9} - \frac{3}{47} a^{7} + \frac{1}{47} a^{3} - \frac{22}{47} a$, $\frac{1}{140083591983670655035080260608} a^{42} + \frac{2256509863799051756369}{6090590955811767610220880896} a^{40} + \frac{2583061964807034038971}{3045295477905883805110440448} a^{38} + \frac{1113260565432830064567}{3045295477905883805110440448} a^{36} - \frac{1447420254003493684027}{1522647738952941902555220224} a^{34} + \frac{25084101878312392539697}{761323869476470951277610112} a^{32} + \frac{47609048433290713788911}{761323869476470951277610112} a^{30} - \frac{76874360248395014426301}{761323869476470951277610112} a^{28} + \frac{30284139663092613672881}{190330967369117737819402528} a^{26} + \frac{4424858096404237967827}{16550518901662411984295872} a^{24} + \frac{4942645013554099176011}{95165483684558868909701264} a^{22} + \frac{139593334965327955092689}{11895685460569858613712658} a^{20} + \frac{526428717228504804087}{517203715676950374509246} a^{18} + \frac{83128699930830570215869}{4137629725415602996073968} a^{16} - \frac{36813317736444162207351}{2068814862707801498036984} a^{14} + \frac{21427888565751798835753}{2068814862707801498036984} a^{12} - \frac{102115646731223828890349}{1034407431353900749018492} a^{10} + \frac{98150372661380921959625}{1034407431353900749018492} a^{8} - \frac{29683296328364497987272}{258601857838475187254623} a^{6} - \frac{46636082839915853335175}{258601857838475187254623} a^{4} + \frac{57343739862383229677412}{258601857838475187254623} a^{2} - \frac{72049589838778838186310}{258601857838475187254623}$, $\frac{1}{140083591983670655035080260608} a^{43} + \frac{2256509863799051756369}{6090590955811767610220880896} a^{41} + \frac{2583061964807034038971}{3045295477905883805110440448} a^{39} + \frac{1113260565432830064567}{3045295477905883805110440448} a^{37} - \frac{1447420254003493684027}{1522647738952941902555220224} a^{35} + \frac{25084101878312392539697}{761323869476470951277610112} a^{33} + \frac{47609048433290713788911}{761323869476470951277610112} a^{31} - \frac{76874360248395014426301}{761323869476470951277610112} a^{29} + \frac{30284139663092613672881}{190330967369117737819402528} a^{27} + \frac{4424858096404237967827}{16550518901662411984295872} a^{25} + \frac{4942645013554099176011}{95165483684558868909701264} a^{23} + \frac{139593334965327955092689}{11895685460569858613712658} a^{21} + \frac{526428717228504804087}{517203715676950374509246} a^{19} + \frac{83128699930830570215869}{4137629725415602996073968} a^{17} - \frac{36813317736444162207351}{2068814862707801498036984} a^{15} + \frac{21427888565751798835753}{2068814862707801498036984} a^{13} - \frac{102115646731223828890349}{1034407431353900749018492} a^{11} + \frac{98150372661380921959625}{1034407431353900749018492} a^{9} - \frac{29683296328364497987272}{258601857838475187254623} a^{7} - \frac{46636082839915853335175}{258601857838475187254623} a^{5} + \frac{57343739862383229677412}{258601857838475187254623} a^{3} - \frac{72049589838778838186310}{258601857838475187254623} a$

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

\(\Q(\sqrt{2}) \), 4.0.1083392.5, \(\Q(\zeta_{23})^+\), 22.22.14741666340843480753092741810452692992.1

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 $44$ $44$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{4}$ $44$ $44$ $22^{2}$ $44$ R $44$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $44$ $22^{2}$ $44$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{44}$ $44$ $44$

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
23Data not computed