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Properties

Label 44.0.860...376.3
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 + 3*x^42 + 9*x^40 + 27*x^38 + 81*x^36 + 243*x^34 + 729*x^32 + 2187*x^30 + 6561*x^28 + 19683*x^26 + 59049*x^24 + 177147*x^22 + 531441*x^20 + 1594323*x^18 + 4782969*x^16 + 14348907*x^14 + 43046721*x^12 + 129140163*x^10 + 387420489*x^8 + 1162261467*x^6 + 3486784401*x^4 + 10460353203*x^2 + 31381059609)
 
gp: K = bnfinit(x^44 + 3*x^42 + 9*x^40 + 27*x^38 + 81*x^36 + 243*x^34 + 729*x^32 + 2187*x^30 + 6561*x^28 + 19683*x^26 + 59049*x^24 + 177147*x^22 + 531441*x^20 + 1594323*x^18 + 4782969*x^16 + 14348907*x^14 + 43046721*x^12 + 129140163*x^10 + 387420489*x^8 + 1162261467*x^6 + 3486784401*x^4 + 10460353203*x^2 + 31381059609, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31381059609, 0, 10460353203, 0, 3486784401, 0, 1162261467, 0, 387420489, 0, 129140163, 0, 43046721, 0, 14348907, 0, 4782969, 0, 1594323, 0, 531441, 0, 177147, 0, 59049, 0, 19683, 0, 6561, 0, 2187, 0, 729, 0, 243, 0, 81, 0, 27, 0, 9, 0, 3, 0, 1]);
 

\( x^{44} + 3 x^{42} + 9 x^{40} + 27 x^{38} + 81 x^{36} + 243 x^{34} + 729 x^{32} + 2187 x^{30} + 6561 x^{28} + 19683 x^{26} + 59049 x^{24} + 177147 x^{22} + 531441 x^{20} + 1594323 x^{18} + 4782969 x^{16} + 14348907 x^{14} + 43046721 x^{12} + 129140163 x^{10} + 387420489 x^{8} + 1162261467 x^{6} + 3486784401 x^{4} + 10460353203 x^{2} + 31381059609 \)

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$$\chi_{276}(1,·)$, $\chi_{276}(131,·)$, $\chi_{276}(133,·)$, $\chi_{276}(263,·)$, $\chi_{276}(265,·)$, $\chi_{276}(11,·)$, $\chi_{276}(13,·)$, $\chi_{276}(143,·)$, $\chi_{276}(145,·)$, $\chi_{276}(275,·)$, $\chi_{276}(25,·)$, $\chi_{276}(155,·)$, $\chi_{276}(157,·)$, $\chi_{276}(35,·)$, $\chi_{276}(37,·)$, $\chi_{276}(167,·)$, $\chi_{276}(169,·)$, $\chi_{276}(47,·)$, $\chi_{276}(49,·)$, $\chi_{276}(179,·)$, $\chi_{276}(181,·)$, $\chi_{276}(59,·)$, $\chi_{276}(61,·)$, $\chi_{276}(191,·)$, $\chi_{276}(193,·)$, $\chi_{276}(71,·)$, $\chi_{276}(73,·)$, $\chi_{276}(203,·)$, $\chi_{276}(205,·)$, $\chi_{276}(83,·)$, $\chi_{276}(85,·)$, $\chi_{276}(215,·)$, $\chi_{276}(217,·)$, $\chi_{276}(95,·)$, $\chi_{276}(97,·)$, $\chi_{276}(227,·)$, $\chi_{276}(229,·)$, $\chi_{276}(107,·)$, $\chi_{276}(109,·)$, $\chi_{276}(239,·)$, $\chi_{276}(241,·)$, $\chi_{276}(119,·)$, $\chi_{276}(121,·)$, $\chi_{276}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{6561} a^{16}$, $\frac{1}{6561} a^{17}$, $\frac{1}{19683} a^{18}$, $\frac{1}{19683} a^{19}$, $\frac{1}{59049} a^{20}$, $\frac{1}{59049} a^{21}$, $\frac{1}{177147} a^{22}$, $\frac{1}{177147} a^{23}$, $\frac{1}{531441} a^{24}$, $\frac{1}{531441} a^{25}$, $\frac{1}{1594323} a^{26}$, $\frac{1}{1594323} a^{27}$, $\frac{1}{4782969} a^{28}$, $\frac{1}{4782969} a^{29}$, $\frac{1}{14348907} a^{30}$, $\frac{1}{14348907} a^{31}$, $\frac{1}{43046721} a^{32}$, $\frac{1}{43046721} a^{33}$, $\frac{1}{129140163} a^{34}$, $\frac{1}{129140163} a^{35}$, $\frac{1}{387420489} a^{36}$, $\frac{1}{387420489} a^{37}$, $\frac{1}{1162261467} a^{38}$, $\frac{1}{1162261467} a^{39}$, $\frac{1}{3486784401} a^{40}$, $\frac{1}{3486784401} a^{41}$, $\frac{1}{10460353203} a^{42}$, $\frac{1}{10460353203} a^{43}$

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:  \( -\frac{1}{387420489} a^{36} \) (order $46$)
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, \(\Q(\zeta_{23})\), 22.22.1275118148086621135238339811277472268288.1

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 $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{44}$ $22^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{4}$

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