Properties

Label 44.0.114...856.2
Degree $44$
Signature $[0, 22]$
Discriminant $1.150\times 10^{77}$
Root discriminant $56.41$
Ramified primes $2, 23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 2*x^42 + 4*x^40 + 8*x^38 + 16*x^36 + 32*x^34 + 64*x^32 + 128*x^30 + 256*x^28 + 512*x^26 + 1024*x^24 + 2048*x^22 + 4096*x^20 + 8192*x^18 + 16384*x^16 + 32768*x^14 + 65536*x^12 + 131072*x^10 + 262144*x^8 + 524288*x^6 + 1048576*x^4 + 2097152*x^2 + 4194304)
 
gp: K = bnfinit(x^44 + 2*x^42 + 4*x^40 + 8*x^38 + 16*x^36 + 32*x^34 + 64*x^32 + 128*x^30 + 256*x^28 + 512*x^26 + 1024*x^24 + 2048*x^22 + 4096*x^20 + 8192*x^18 + 16384*x^16 + 32768*x^14 + 65536*x^12 + 131072*x^10 + 262144*x^8 + 524288*x^6 + 1048576*x^4 + 2097152*x^2 + 4194304, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4194304, 0, 2097152, 0, 1048576, 0, 524288, 0, 262144, 0, 131072, 0, 65536, 0, 32768, 0, 16384, 0, 8192, 0, 4096, 0, 2048, 0, 1024, 0, 512, 0, 256, 0, 128, 0, 64, 0, 32, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1]);
 

\( x^{44} + 2 x^{42} + 4 x^{40} + 8 x^{38} + 16 x^{36} + 32 x^{34} + 64 x^{32} + 128 x^{30} + 256 x^{28} + 512 x^{26} + 1024 x^{24} + 2048 x^{22} + 4096 x^{20} + 8192 x^{18} + 16384 x^{16} + 32768 x^{14} + 65536 x^{12} + 131072 x^{10} + 262144 x^{8} + 524288 x^{6} + 1048576 x^{4} + 2097152 x^{2} + 4194304 \)

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:  \(114\!\cdots\!856\)\(\medspace = 2^{66}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $56.41$
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$$\chi_{184}(1,·)$, $\chi_{184}(5,·)$, $\chi_{184}(129,·)$, $\chi_{184}(9,·)$, $\chi_{184}(13,·)$, $\chi_{184}(45,·)$, $\chi_{184}(17,·)$, $\chi_{184}(21,·)$, $\chi_{184}(153,·)$, $\chi_{184}(25,·)$, $\chi_{184}(157,·)$, $\chi_{184}(133,·)$, $\chi_{184}(33,·)$, $\chi_{184}(37,·)$, $\chi_{184}(49,·)$, $\chi_{184}(41,·)$, $\chi_{184}(173,·)$, $\chi_{184}(29,·)$, $\chi_{184}(177,·)$, $\chi_{184}(53,·)$, $\chi_{184}(137,·)$, $\chi_{184}(57,·)$, $\chi_{184}(61,·)$, $\chi_{184}(181,·)$, $\chi_{184}(65,·)$, $\chi_{184}(73,·)$, $\chi_{184}(77,·)$, $\chi_{184}(141,·)$, $\chi_{184}(81,·)$, $\chi_{184}(85,·)$, $\chi_{184}(89,·)$, $\chi_{184}(93,·)$, $\chi_{184}(165,·)$, $\chi_{184}(97,·)$, $\chi_{184}(101,·)$, $\chi_{184}(145,·)$, $\chi_{184}(105,·)$, $\chi_{184}(109,·)$, $\chi_{184}(113,·)$, $\chi_{184}(117,·)$, $\chi_{184}(169,·)$, $\chi_{184}(121,·)$, $\chi_{184}(125,·)$, $\chi_{184}(149,·)$$\rbrace$
This is a CM field.

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

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

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:  \( -\frac{1}{2} a^{2} \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{2}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.22.14741666340843480753092741810452692992.1, \(\Q(\zeta_{23})\), 22.0.339058325839400057321133061640411938816.1

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 $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{44}$ $22^{2}$ $22^{2}$

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