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Properties

Label 44.0.36075838195...4304.1
Degree $44$
Signature $[0, 22]$
Discriminant $2^{66}\cdot 3^{22}\cdot 23^{42}$
Root discriminant $97.71$
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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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4194304, 0, 369098752, 0, 8074035200, 0, 69840404480, 0, 318866718720, 0, 888799887360, 0, 1650124062720, 0, 2160531210240, 0, 2075463352320, 0, 1504631316480, 0, 840073850880, 0, 366451783680, 0, 126096215040, 0, 34415527424, 0, 7462070016, 0, 1282144128, 0, 173330496, 0, 18195712, 0, 1451584, 0, 84976, 0, 3440, 0, 86, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 86*x^42 + 3440*x^40 + 84976*x^38 + 1451584*x^36 + 18195712*x^34 + 173330496*x^32 + 1282144128*x^30 + 7462070016*x^28 + 34415527424*x^26 + 126096215040*x^24 + 366451783680*x^22 + 840073850880*x^20 + 1504631316480*x^18 + 2075463352320*x^16 + 2160531210240*x^14 + 1650124062720*x^12 + 888799887360*x^10 + 318866718720*x^8 + 69840404480*x^6 + 8074035200*x^4 + 369098752*x^2 + 4194304)
 
gp: K = bnfinit(x^44 + 86*x^42 + 3440*x^40 + 84976*x^38 + 1451584*x^36 + 18195712*x^34 + 173330496*x^32 + 1282144128*x^30 + 7462070016*x^28 + 34415527424*x^26 + 126096215040*x^24 + 366451783680*x^22 + 840073850880*x^20 + 1504631316480*x^18 + 2075463352320*x^16 + 2160531210240*x^14 + 1650124062720*x^12 + 888799887360*x^10 + 318866718720*x^8 + 69840404480*x^6 + 8074035200*x^4 + 369098752*x^2 + 4194304, 1)
 

Normalized defining polynomial

\( x^{44} + 86 x^{42} + 3440 x^{40} + 84976 x^{38} + 1451584 x^{36} + 18195712 x^{34} + 173330496 x^{32} + 1282144128 x^{30} + 7462070016 x^{28} + 34415527424 x^{26} + 126096215040 x^{24} + 366451783680 x^{22} + 840073850880 x^{20} + 1504631316480 x^{18} + 2075463352320 x^{16} + 2160531210240 x^{14} + 1650124062720 x^{12} + 888799887360 x^{10} + 318866718720 x^{8} + 69840404480 x^{6} + 8074035200 x^{4} + 369098752 x^{2} + 4194304 \)

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:  \(3607583819545152459027384276140645884702253651640471494457079112798455926939134201954304=2^{66}\cdot 3^{22}\cdot 23^{42}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.71$
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:  $2, 3, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(552=2^{3}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{552}(1,·)$, $\chi_{552}(517,·)$, $\chi_{552}(65,·)$, $\chi_{552}(521,·)$, $\chi_{552}(269,·)$, $\chi_{552}(109,·)$, $\chi_{552}(17,·)$, $\chi_{552}(205,·)$, $\chi_{552}(533,·)$, $\chi_{552}(73,·)$, $\chi_{552}(281,·)$, $\chi_{552}(409,·)$, $\chi_{552}(265,·)$, $\chi_{552}(29,·)$, $\chi_{552}(289,·)$, $\chi_{552}(37,·)$, $\chi_{552}(113,·)$, $\chi_{552}(169,·)$, $\chi_{552}(173,·)$, $\chi_{552}(157,·)$, $\chi_{552}(49,·)$, $\chi_{552}(181,·)$, $\chi_{552}(137,·)$, $\chi_{552}(317,·)$, $\chi_{552}(193,·)$, $\chi_{552}(197,·)$, $\chi_{552}(329,·)$, $\chi_{552}(461,·)$, $\chi_{552}(77,·)$, $\chi_{552}(89,·)$, $\chi_{552}(101,·)$, $\chi_{552}(421,·)$, $\chi_{552}(229,·)$, $\chi_{552}(401,·)$, $\chi_{552}(361,·)$, $\chi_{552}(485,·)$, $\chi_{552}(493,·)$, $\chi_{552}(61,·)$, $\chi_{552}(497,·)$, $\chi_{552}(25,·)$, $\chi_{552}(373,·)$, $\chi_{552}(425,·)$, $\chi_{552}(121,·)$, $\chi_{552}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$, $\frac{1}{262144} a^{36}$, $\frac{1}{262144} a^{37}$, $\frac{1}{524288} a^{38}$, $\frac{1}{524288} a^{39}$, $\frac{1}{1048576} a^{40}$, $\frac{1}{1048576} a^{41}$, $\frac{1}{2097152} a^{42}$, $\frac{1}{2097152} a^{43}$

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

$C_2\times C_{22}$ (as 44T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{69}) \), \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-6}, \sqrt{-46})\), \(\Q(\zeta_{23})^+\), \(\Q(\zeta_{69})^+\), 22.0.339058325839400057321133061640411938816.1, 22.0.2611441967281400084968119933496263205453824.1

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 ${\href{/LocalNumberField/5.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{4}$ $22^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
3Data not computed
23Data not computed