Properties

Label 44.44.860...376.1
Degree $44$
Signature $[44, 0]$
Discriminant $8.601\times 10^{80}$
Root discriminant $69.09$
Ramified primes $2, 3, 23$
Class number $1$ (GRH)
Class group trivial (GRH)
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 - 43*x^42 + 860*x^40 - 10622*x^38 + 90724*x^36 - 568616*x^34 + 2708289*x^32 - 10016751*x^30 + 29148711*x^28 - 67217827*x^26 + 123140835*x^24 - 178931535*x^22 + 205096155*x^20 - 183670815*x^18 + 126676230*x^16 - 65934180*x^14 + 25178895*x^12 - 6781005*x^10 + 1216380*x^8 - 133210*x^6 + 7700*x^4 - 176*x^2 + 1)
 
gp: K = bnfinit(x^44 - 43*x^42 + 860*x^40 - 10622*x^38 + 90724*x^36 - 568616*x^34 + 2708289*x^32 - 10016751*x^30 + 29148711*x^28 - 67217827*x^26 + 123140835*x^24 - 178931535*x^22 + 205096155*x^20 - 183670815*x^18 + 126676230*x^16 - 65934180*x^14 + 25178895*x^12 - 6781005*x^10 + 1216380*x^8 - 133210*x^6 + 7700*x^4 - 176*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -176, 0, 7700, 0, -133210, 0, 1216380, 0, -6781005, 0, 25178895, 0, -65934180, 0, 126676230, 0, -183670815, 0, 205096155, 0, -178931535, 0, 123140835, 0, -67217827, 0, 29148711, 0, -10016751, 0, 2708289, 0, -568616, 0, 90724, 0, -10622, 0, 860, 0, -43, 0, 1]);
 

\( x^{44} - 43 x^{42} + 860 x^{40} - 10622 x^{38} + 90724 x^{36} - 568616 x^{34} + 2708289 x^{32} - 10016751 x^{30} + 29148711 x^{28} - 67217827 x^{26} + 123140835 x^{24} - 178931535 x^{22} + 205096155 x^{20} - 183670815 x^{18} + 126676230 x^{16} - 65934180 x^{14} + 25178895 x^{12} - 6781005 x^{10} + 1216380 x^{8} - 133210 x^{6} + 7700 x^{4} - 176 x^{2} + 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[44, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(860\!\cdots\!376\)\(\medspace = 2^{44}\cdot 3^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $69.09$
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:  $2, 3, 23$
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:  \(276=2^{2}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{276}(1,·)$, $\chi_{276}(131,·)$, $\chi_{276}(5,·)$, $\chi_{276}(7,·)$, $\chi_{276}(265,·)$, $\chi_{276}(13,·)$, $\chi_{276}(17,·)$, $\chi_{276}(19,·)$, $\chi_{276}(149,·)$, $\chi_{276}(25,·)$, $\chi_{276}(47,·)$, $\chi_{276}(133,·)$, $\chi_{276}(35,·)$, $\chi_{276}(65,·)$, $\chi_{276}(167,·)$, $\chi_{276}(169,·)$, $\chi_{276}(43,·)$, $\chi_{276}(175,·)$, $\chi_{276}(49,·)$, $\chi_{276}(179,·)$, $\chi_{276}(53,·)$, $\chi_{276}(137,·)$, $\chi_{276}(59,·)$, $\chi_{276}(193,·)$, $\chi_{276}(67,·)$, $\chi_{276}(71,·)$, $\chi_{276}(73,·)$, $\chi_{276}(119,·)$, $\chi_{276}(79,·)$, $\chi_{276}(85,·)$, $\chi_{276}(215,·)$, $\chi_{276}(89,·)$, $\chi_{276}(91,·)$, $\chi_{276}(221,·)$, $\chi_{276}(199,·)$, $\chi_{276}(95,·)$, $\chi_{276}(103,·)$, $\chi_{276}(235,·)$, $\chi_{276}(239,·)$, $\chi_{276}(113,·)$, $\chi_{276}(245,·)$, $\chi_{276}(247,·)$, $\chi_{276}(121,·)$, $\chi_{276}(125,·)$$\rbrace$
This is not 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

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $43$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 396826934994876900000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{44}\cdot(2\pi)^{0}\cdot 396826934994876900000000000 \cdot 1}{2\sqrt{860115008245742907292219227824365111518443501386754869093198564719785672888549376}}\approx 0.119018012401450$ (assuming GRH)

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

\(\Q(\sqrt{69}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{3}, \sqrt{23})\), \(\Q(\zeta_{23})^+\), \(\Q(\zeta_{69})^+\), \(\Q(\zeta_{92})^+\), 22.22.1275118148086621135238339811277472268288.1

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 $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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
3Data not computed
23Data not computed