Properties

Label 44.0.653...000.3
Degree $44$
Signature $[0, 22]$
Discriminant $6.535\times 10^{85}$
Root discriminant $89.20$
Ramified primes $2, 5, 23$
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 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625)
 
gp: K = bnfinit(x^44 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2384185791015625, 0, -476837158203125, 0, 95367431640625, 0, -19073486328125, 0, 3814697265625, 0, -762939453125, 0, 152587890625, 0, -30517578125, 0, 6103515625, 0, -1220703125, 0, 244140625, 0, -48828125, 0, 9765625, 0, -1953125, 0, 390625, 0, -78125, 0, 15625, 0, -3125, 0, 625, 0, -125, 0, 25, 0, -5, 0, 1]);
 

\( x^{44} - 5 x^{42} + 25 x^{40} - 125 x^{38} + 625 x^{36} - 3125 x^{34} + 15625 x^{32} - 78125 x^{30} + 390625 x^{28} - 1953125 x^{26} + 9765625 x^{24} - 48828125 x^{22} + 244140625 x^{20} - 1220703125 x^{18} + 6103515625 x^{16} - 30517578125 x^{14} + 152587890625 x^{12} - 762939453125 x^{10} + 3814697265625 x^{8} - 19073486328125 x^{6} + 95367431640625 x^{4} - 476837158203125 x^{2} + 2384185791015625 \)

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:  \(653\!\cdots\!000\)\(\medspace = 2^{44}\cdot 5^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $89.20$
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:  $2, 5, 23$
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:  \(460=2^{2}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{460}(1,·)$, $\chi_{460}(259,·)$, $\chi_{460}(261,·)$, $\chi_{460}(139,·)$, $\chi_{460}(141,·)$, $\chi_{460}(399,·)$, $\chi_{460}(401,·)$, $\chi_{460}(19,·)$, $\chi_{460}(21,·)$, $\chi_{460}(279,·)$, $\chi_{460}(281,·)$, $\chi_{460}(159,·)$, $\chi_{460}(419,·)$, $\chi_{460}(421,·)$, $\chi_{460}(39,·)$, $\chi_{460}(41,·)$, $\chi_{460}(301,·)$, $\chi_{460}(179,·)$, $\chi_{460}(181,·)$, $\chi_{460}(439,·)$, $\chi_{460}(441,·)$, $\chi_{460}(59,·)$, $\chi_{460}(61,·)$, $\chi_{460}(319,·)$, $\chi_{460}(321,·)$, $\chi_{460}(199,·)$, $\chi_{460}(201,·)$, $\chi_{460}(459,·)$, $\chi_{460}(79,·)$, $\chi_{460}(81,·)$, $\chi_{460}(339,·)$, $\chi_{460}(341,·)$, $\chi_{460}(219,·)$, $\chi_{460}(221,·)$, $\chi_{460}(99,·)$, $\chi_{460}(101,·)$, $\chi_{460}(359,·)$, $\chi_{460}(361,·)$, $\chi_{460}(239,·)$, $\chi_{460}(241,·)$, $\chi_{460}(119,·)$, $\chi_{460}(121,·)$, $\chi_{460}(379,·)$, $\chi_{460}(381,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$, $\frac{1}{625} a^{8}$, $\frac{1}{625} a^{9}$, $\frac{1}{3125} a^{10}$, $\frac{1}{3125} a^{11}$, $\frac{1}{15625} a^{12}$, $\frac{1}{15625} a^{13}$, $\frac{1}{78125} a^{14}$, $\frac{1}{78125} a^{15}$, $\frac{1}{390625} a^{16}$, $\frac{1}{390625} a^{17}$, $\frac{1}{1953125} a^{18}$, $\frac{1}{1953125} a^{19}$, $\frac{1}{9765625} a^{20}$, $\frac{1}{9765625} a^{21}$, $\frac{1}{48828125} a^{22}$, $\frac{1}{48828125} a^{23}$, $\frac{1}{244140625} a^{24}$, $\frac{1}{244140625} a^{25}$, $\frac{1}{1220703125} a^{26}$, $\frac{1}{1220703125} a^{27}$, $\frac{1}{6103515625} a^{28}$, $\frac{1}{6103515625} a^{29}$, $\frac{1}{30517578125} a^{30}$, $\frac{1}{30517578125} a^{31}$, $\frac{1}{152587890625} a^{32}$, $\frac{1}{152587890625} a^{33}$, $\frac{1}{762939453125} a^{34}$, $\frac{1}{762939453125} a^{35}$, $\frac{1}{3814697265625} a^{36}$, $\frac{1}{3814697265625} a^{37}$, $\frac{1}{19073486328125} a^{38}$, $\frac{1}{19073486328125} a^{39}$, $\frac{1}{95367431640625} a^{40}$, $\frac{1}{95367431640625} a^{41}$, $\frac{1}{476837158203125} a^{42}$, $\frac{1}{476837158203125} a^{43}$

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:  \( \frac{1}{3125} a^{10} \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{115}) \), \(\Q(\sqrt{-5}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.0.351468714257323283030813737164800000000000.1, \(\Q(\zeta_{23})\), 22.22.8083780427918435509708715954790400000000000.1

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 ${\href{/LocalNumberField/3.11.0.1}{11} }^{4}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{44}$ $22^{2}$ $22^{2}$

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
5Data not computed
23Data not computed