Properties

Label 44.0.482...224.1
Degree $44$
Signature $[0, 22]$
Discriminant $4.822\times 10^{83}$
Root discriminant $79.78$
Ramified primes $2, 23$
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 + 44*x^42 + 901*x^40 + 11400*x^38 + 99790*x^36 + 641208*x^34 + 3131721*x^32 + 11878176*x^30 + 35442612*x^28 + 83778736*x^26 + 157236844*x^24 + 233880352*x^22 + 274130056*x^20 + 250699168*x^18 + 176290339*x^16 + 93382192*x^14 + 36217051*x^12 + 9883588*x^10 + 1792219*x^8 + 197912*x^6 + 11506*x^4 + 264*x^2 + 1)
 
gp: K = bnfinit(x^44 + 44*x^42 + 901*x^40 + 11400*x^38 + 99790*x^36 + 641208*x^34 + 3131721*x^32 + 11878176*x^30 + 35442612*x^28 + 83778736*x^26 + 157236844*x^24 + 233880352*x^22 + 274130056*x^20 + 250699168*x^18 + 176290339*x^16 + 93382192*x^14 + 36217051*x^12 + 9883588*x^10 + 1792219*x^8 + 197912*x^6 + 11506*x^4 + 264*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 264, 0, 11506, 0, 197912, 0, 1792219, 0, 9883588, 0, 36217051, 0, 93382192, 0, 176290339, 0, 250699168, 0, 274130056, 0, 233880352, 0, 157236844, 0, 83778736, 0, 35442612, 0, 11878176, 0, 3131721, 0, 641208, 0, 99790, 0, 11400, 0, 901, 0, 44, 0, 1]);
 

\( x^{44} + 44 x^{42} + 901 x^{40} + 11400 x^{38} + 99790 x^{36} + 641208 x^{34} + 3131721 x^{32} + 11878176 x^{30} + 35442612 x^{28} + 83778736 x^{26} + 157236844 x^{24} + 233880352 x^{22} + 274130056 x^{20} + 250699168 x^{18} + 176290339 x^{16} + 93382192 x^{14} + 36217051 x^{12} + 9883588 x^{10} + 1792219 x^{8} + 197912 x^{6} + 11506 x^{4} + 264 x^{2} + 1 \)

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:  \(482\!\cdots\!224\)\(\medspace = 2^{88}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $79.78$
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:  $2, 23$
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:  \(184=2^{3}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{184}(1,·)$, $\chi_{184}(3,·)$, $\chi_{184}(5,·)$, $\chi_{184}(7,·)$, $\chi_{184}(9,·)$, $\chi_{184}(139,·)$, $\chi_{184}(15,·)$, $\chi_{184}(131,·)$, $\chi_{184}(21,·)$, $\chi_{184}(25,·)$, $\chi_{184}(27,·)$, $\chi_{184}(157,·)$, $\chi_{184}(159,·)$, $\chi_{184}(35,·)$, $\chi_{184}(37,·)$, $\chi_{184}(177,·)$, $\chi_{184}(41,·)$, $\chi_{184}(135,·)$, $\chi_{184}(45,·)$, $\chi_{184}(175,·)$, $\chi_{184}(49,·)$, $\chi_{184}(179,·)$, $\chi_{184}(53,·)$, $\chi_{184}(183,·)$, $\chi_{184}(59,·)$, $\chi_{184}(61,·)$, $\chi_{184}(63,·)$, $\chi_{184}(181,·)$, $\chi_{184}(73,·)$, $\chi_{184}(75,·)$, $\chi_{184}(79,·)$, $\chi_{184}(81,·)$, $\chi_{184}(163,·)$, $\chi_{184}(143,·)$, $\chi_{184}(103,·)$, $\chi_{184}(105,·)$, $\chi_{184}(109,·)$, $\chi_{184}(111,·)$, $\chi_{184}(147,·)$, $\chi_{184}(169,·)$, $\chi_{184}(121,·)$, $\chi_{184}(123,·)$, $\chi_{184}(125,·)$, $\chi_{184}(149,·)$$\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:  \( -1 \) (order $2$)
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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{-2}, \sqrt{23})\), \(\Q(\zeta_{23})^+\), 22.0.14741666340843480753092741810452692992.1, 22.0.339058325839400057321133061640411938816.1, \(\Q(\zeta_{92})^+\)

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 $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{4}$ R $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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
23Data not computed