Properties

Label 44.0.371...625.1
Degree $44$
Signature $[0, 22]$
Discriminant $3.715\times 10^{72}$
Root discriminant $44.60$
Ramified primes $5, 23$
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 - x^43 + 2*x^42 - 3*x^41 + 5*x^40 - 8*x^39 + 13*x^38 - 21*x^37 + 34*x^36 - 55*x^35 + 89*x^34 - 144*x^33 + 233*x^32 - 377*x^31 + 610*x^30 - 987*x^29 + 1597*x^28 - 2584*x^27 + 4181*x^26 - 6765*x^25 + 10946*x^24 - 17711*x^23 + 28657*x^22 + 17711*x^21 + 10946*x^20 + 6765*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 
gp: K = bnfinit(x^44 - x^43 + 2*x^42 - 3*x^41 + 5*x^40 - 8*x^39 + 13*x^38 - 21*x^37 + 34*x^36 - 55*x^35 + 89*x^34 - 144*x^33 + 233*x^32 - 377*x^31 + 610*x^30 - 987*x^29 + 1597*x^28 - 2584*x^27 + 4181*x^26 - 6765*x^25 + 10946*x^24 - 17711*x^23 + 28657*x^22 + 17711*x^21 + 10946*x^20 + 6765*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, -17711, 10946, -6765, 4181, -2584, 1597, -987, 610, -377, 233, -144, 89, -55, 34, -21, 13, -8, 5, -3, 2, -1, 1]);
 

\( x^{44} - x^{43} + 2 x^{42} - 3 x^{41} + 5 x^{40} - 8 x^{39} + 13 x^{38} - 21 x^{37} + 34 x^{36} - 55 x^{35} + 89 x^{34} - 144 x^{33} + 233 x^{32} - 377 x^{31} + 610 x^{30} - 987 x^{29} + 1597 x^{28} - 2584 x^{27} + 4181 x^{26} - 6765 x^{25} + 10946 x^{24} - 17711 x^{23} + 28657 x^{22} + 17711 x^{21} + 10946 x^{20} + 6765 x^{19} + 4181 x^{18} + 2584 x^{17} + 1597 x^{16} + 987 x^{15} + 610 x^{14} + 377 x^{13} + 233 x^{12} + 144 x^{11} + 89 x^{10} + 55 x^{9} + 34 x^{8} + 21 x^{7} + 13 x^{6} + 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1 \)

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:  \(371\!\cdots\!625\)\(\medspace = 5^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $44.60$
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:  $5, 23$
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:  \(115=5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{115}(1,·)$, $\chi_{115}(4,·)$, $\chi_{115}(6,·)$, $\chi_{115}(9,·)$, $\chi_{115}(11,·)$, $\chi_{115}(14,·)$, $\chi_{115}(16,·)$, $\chi_{115}(19,·)$, $\chi_{115}(21,·)$, $\chi_{115}(24,·)$, $\chi_{115}(26,·)$, $\chi_{115}(29,·)$, $\chi_{115}(31,·)$, $\chi_{115}(34,·)$, $\chi_{115}(36,·)$, $\chi_{115}(39,·)$, $\chi_{115}(41,·)$, $\chi_{115}(44,·)$, $\chi_{115}(49,·)$, $\chi_{115}(51,·)$, $\chi_{115}(54,·)$, $\chi_{115}(56,·)$, $\chi_{115}(59,·)$, $\chi_{115}(61,·)$, $\chi_{115}(64,·)$, $\chi_{115}(66,·)$, $\chi_{115}(71,·)$, $\chi_{115}(74,·)$, $\chi_{115}(76,·)$, $\chi_{115}(79,·)$, $\chi_{115}(81,·)$, $\chi_{115}(84,·)$, $\chi_{115}(86,·)$, $\chi_{115}(89,·)$, $\chi_{115}(91,·)$, $\chi_{115}(94,·)$, $\chi_{115}(96,·)$, $\chi_{115}(99,·)$, $\chi_{115}(101,·)$, $\chi_{115}(104,·)$, $\chi_{115}(106,·)$, $\chi_{115}(109,·)$, $\chi_{115}(111,·)$, $\chi_{115}(114,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{28657} a^{23} - \frac{10946}{28657}$, $\frac{1}{28657} a^{24} - \frac{10946}{28657} a$, $\frac{1}{28657} a^{25} - \frac{10946}{28657} a^{2}$, $\frac{1}{28657} a^{26} - \frac{10946}{28657} a^{3}$, $\frac{1}{28657} a^{27} - \frac{10946}{28657} a^{4}$, $\frac{1}{28657} a^{28} - \frac{10946}{28657} a^{5}$, $\frac{1}{28657} a^{29} - \frac{10946}{28657} a^{6}$, $\frac{1}{28657} a^{30} - \frac{10946}{28657} a^{7}$, $\frac{1}{28657} a^{31} - \frac{10946}{28657} a^{8}$, $\frac{1}{28657} a^{32} - \frac{10946}{28657} a^{9}$, $\frac{1}{28657} a^{33} - \frac{10946}{28657} a^{10}$, $\frac{1}{28657} a^{34} - \frac{10946}{28657} a^{11}$, $\frac{1}{28657} a^{35} - \frac{10946}{28657} a^{12}$, $\frac{1}{28657} a^{36} - \frac{10946}{28657} a^{13}$, $\frac{1}{28657} a^{37} - \frac{10946}{28657} a^{14}$, $\frac{1}{28657} a^{38} - \frac{10946}{28657} a^{15}$, $\frac{1}{28657} a^{39} - \frac{10946}{28657} a^{16}$, $\frac{1}{28657} a^{40} - \frac{10946}{28657} a^{17}$, $\frac{1}{28657} a^{41} - \frac{10946}{28657} a^{18}$, $\frac{1}{28657} a^{42} - \frac{10946}{28657} a^{19}$, $\frac{1}{28657} a^{43} - \frac{10946}{28657} a^{20}$

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:  \( -\frac{610}{28657} a^{38} - \frac{39088169}{28657} a^{15} \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{5}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1, \(\Q(\zeta_{23})\), 22.0.1927323443393334271838358868310546875.1

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 $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{4}$

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
5Data not computed
23Data not computed