Properties

Label 44.0.860...376.1
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 + 45*x^42 + 944*x^40 + 12258*x^38 + 110332*x^36 + 730456*x^34 + 3683681*x^32 + 14457937*x^30 + 44741027*x^28 + 109922933*x^26 + 214871383*x^24 + 333482657*x^22 + 408408103*x^20 + 390551177*x^18 + 287137498*x^16 + 158825372*x^14 + 64156063*x^12 + 18161627*x^10 + 3397768*x^8 + 384582*x^6 + 22748*x^4 + 528*x^2 + 1)
 
gp: K = bnfinit(x^44 + 45*x^42 + 944*x^40 + 12258*x^38 + 110332*x^36 + 730456*x^34 + 3683681*x^32 + 14457937*x^30 + 44741027*x^28 + 109922933*x^26 + 214871383*x^24 + 333482657*x^22 + 408408103*x^20 + 390551177*x^18 + 287137498*x^16 + 158825372*x^14 + 64156063*x^12 + 18161627*x^10 + 3397768*x^8 + 384582*x^6 + 22748*x^4 + 528*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 528, 0, 22748, 0, 384582, 0, 3397768, 0, 18161627, 0, 64156063, 0, 158825372, 0, 287137498, 0, 390551177, 0, 408408103, 0, 333482657, 0, 214871383, 0, 109922933, 0, 44741027, 0, 14457937, 0, 3683681, 0, 730456, 0, 110332, 0, 12258, 0, 944, 0, 45, 0, 1]);
 

\( x^{44} + 45 x^{42} + 944 x^{40} + 12258 x^{38} + 110332 x^{36} + 730456 x^{34} + 3683681 x^{32} + 14457937 x^{30} + 44741027 x^{28} + 109922933 x^{26} + 214871383 x^{24} + 333482657 x^{22} + 408408103 x^{20} + 390551177 x^{18} + 287137498 x^{16} + 158825372 x^{14} + 64156063 x^{12} + 18161627 x^{10} + 3397768 x^{8} + 384582 x^{6} + 22748 x^{4} + 528 x^{2} + 1 \)

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:  \(860\!\cdots\!376\)\(\medspace = 2^{44}\cdot 3^{22}\cdot 23^{42}\)
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$$\chi_{276}(1,·)$, $\chi_{276}(259,·)$, $\chi_{276}(133,·)$, $\chi_{276}(263,·)$, $\chi_{276}(137,·)$, $\chi_{276}(11,·)$, $\chi_{276}(13,·)$, $\chi_{276}(143,·)$, $\chi_{276}(17,·)$, $\chi_{276}(275,·)$, $\chi_{276}(149,·)$, $\chi_{276}(151,·)$, $\chi_{276}(25,·)$, $\chi_{276}(155,·)$, $\chi_{276}(5,·)$, $\chi_{276}(163,·)$, $\chi_{276}(65,·)$, $\chi_{276}(227,·)$, $\chi_{276}(169,·)$, $\chi_{276}(49,·)$, $\chi_{276}(53,·)$, $\chi_{276}(265,·)$, $\chi_{276}(31,·)$, $\chi_{276}(191,·)$, $\chi_{276}(193,·)$, $\chi_{276}(139,·)$, $\chi_{276}(73,·)$, $\chi_{276}(55,·)$, $\chi_{276}(83,·)$, $\chi_{276}(85,·)$, $\chi_{276}(203,·)$, $\chi_{276}(89,·)$, $\chi_{276}(271,·)$, $\chi_{276}(221,·)$, $\chi_{276}(223,·)$, $\chi_{276}(187,·)$, $\chi_{276}(107,·)$, $\chi_{276}(113,·)$, $\chi_{276}(211,·)$, $\chi_{276}(245,·)$, $\chi_{276}(121,·)$, $\chi_{276}(251,·)$, $\chi_{276}(125,·)$, $\chi_{276}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$

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:  \( a^{23} + 23 a^{21} + 230 a^{19} + 1311 a^{17} + 4692 a^{15} + 10948 a^{13} + 16744 a^{11} + 16445 a^{9} + 9867 a^{7} + 3289 a^{5} + 506 a^{3} + 23 a \) (order $4$)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{-69}) \), \(\Q(i, \sqrt{69})\), \(\Q(\zeta_{23})^+\), 22.0.7198079267989980836471065337135104.1, \(\Q(\zeta_{69})^+\), 22.0.29327717405992286110481815659381862170624.1

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 ${\href{/LocalNumberField/5.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{4}$ $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{4}$ $22^{2}$

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