Properties

Label 44.0.482...224.2
Degree $44$
Signature $[0, 22]$
Discriminant $4.822\times 10^{83}$
Root discriminant $79.78$
Ramified primes $2, 23$
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 + 69*x^40 + 1978*x^36 + 30521*x^32 + 274712*x^28 + 1463260*x^24 + 4481688*x^20 + 7339875*x^16 + 5623799*x^12 + 1588587*x^8 + 104742*x^4 + 529)
 
gp: K = bnfinit(x^44 + 69*x^40 + 1978*x^36 + 30521*x^32 + 274712*x^28 + 1463260*x^24 + 4481688*x^20 + 7339875*x^16 + 5623799*x^12 + 1588587*x^8 + 104742*x^4 + 529, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, 0, 0, 0, 104742, 0, 0, 0, 1588587, 0, 0, 0, 5623799, 0, 0, 0, 7339875, 0, 0, 0, 4481688, 0, 0, 0, 1463260, 0, 0, 0, 274712, 0, 0, 0, 30521, 0, 0, 0, 1978, 0, 0, 0, 69, 0, 0, 0, 1]);
 

\(x^{44} + 69 x^{40} + 1978 x^{36} + 30521 x^{32} + 274712 x^{28} + 1463260 x^{24} + 4481688 x^{20} + 7339875 x^{16} + 5623799 x^{12} + 1588587 x^{8} + 104742 x^{4} + 529\)  Toggle raw display

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:  \(482\!\cdots\!224\)\(\medspace = 2^{88}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $79.78$
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$$\chi_{184}(1,·)$, $\chi_{184}(171,·)$, $\chi_{184}(5,·)$, $\chi_{184}(9,·)$, $\chi_{184}(11,·)$, $\chi_{184}(19,·)$, $\chi_{184}(149,·)$, $\chi_{184}(151,·)$, $\chi_{184}(25,·)$, $\chi_{184}(155,·)$, $\chi_{184}(157,·)$, $\chi_{184}(31,·)$, $\chi_{184}(37,·)$, $\chi_{184}(167,·)$, $\chi_{184}(41,·)$, $\chi_{184}(43,·)$, $\chi_{184}(45,·)$, $\chi_{184}(47,·)$, $\chi_{184}(49,·)$, $\chi_{184}(51,·)$, $\chi_{184}(53,·)$, $\chi_{184}(55,·)$, $\chi_{184}(61,·)$, $\chi_{184}(181,·)$, $\chi_{184}(67,·)$, $\chi_{184}(71,·)$, $\chi_{184}(73,·)$, $\chi_{184}(119,·)$, $\chi_{184}(81,·)$, $\chi_{184}(83,·)$, $\chi_{184}(87,·)$, $\chi_{184}(91,·)$, $\chi_{184}(95,·)$, $\chi_{184}(99,·)$, $\chi_{184}(105,·)$, $\chi_{184}(107,·)$, $\chi_{184}(109,·)$, $\chi_{184}(177,·)$, $\chi_{184}(169,·)$, $\chi_{184}(121,·)$, $\chi_{184}(127,·)$, $\chi_{184}(125,·)$, $\chi_{184}(39,·)$, $\chi_{184}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23} a^{22}$, $\frac{1}{23} a^{23}$, $\frac{1}{23} a^{24}$, $\frac{1}{23} a^{25}$, $\frac{1}{23} a^{26}$, $\frac{1}{23} a^{27}$, $\frac{1}{23} a^{28}$, $\frac{1}{23} a^{29}$, $\frac{1}{23} a^{30}$, $\frac{1}{23} a^{31}$, $\frac{1}{23} a^{32}$, $\frac{1}{23} a^{33}$, $\frac{1}{23} a^{34}$, $\frac{1}{23} a^{35}$, $\frac{1}{23} a^{36}$, $\frac{1}{23} a^{37}$, $\frac{1}{23} a^{38}$, $\frac{1}{23} a^{39}$, $\frac{1}{20449419753638432679768691} a^{40} - \frac{377495412904320220827686}{20449419753638432679768691} a^{36} - \frac{61573360947580349157897}{20449419753638432679768691} a^{32} + \frac{313099305151942484946038}{20449419753638432679768691} a^{28} - \frac{271525895659264597164129}{20449419753638432679768691} a^{24} - \frac{94271963813006501588424}{889105206679931855642117} a^{20} + \frac{160260204316967552982876}{889105206679931855642117} a^{16} + \frac{352251185726095770867734}{889105206679931855642117} a^{12} + \frac{37292798999275391062173}{889105206679931855642117} a^{8} - \frac{269364708807597326628552}{889105206679931855642117} a^{4} - \frac{1747256001144836141445}{889105206679931855642117}$, $\frac{1}{20449419753638432679768691} a^{41} - \frac{377495412904320220827686}{20449419753638432679768691} a^{37} - \frac{61573360947580349157897}{20449419753638432679768691} a^{33} + \frac{313099305151942484946038}{20449419753638432679768691} a^{29} - \frac{271525895659264597164129}{20449419753638432679768691} a^{25} - \frac{94271963813006501588424}{889105206679931855642117} a^{21} + \frac{160260204316967552982876}{889105206679931855642117} a^{17} + \frac{352251185726095770867734}{889105206679931855642117} a^{13} + \frac{37292798999275391062173}{889105206679931855642117} a^{9} - \frac{269364708807597326628552}{889105206679931855642117} a^{5} - \frac{1747256001144836141445}{889105206679931855642117} a$, $\frac{1}{20449419753638432679768691} a^{42} - \frac{377495412904320220827686}{20449419753638432679768691} a^{38} - \frac{61573360947580349157897}{20449419753638432679768691} a^{34} + \frac{313099305151942484946038}{20449419753638432679768691} a^{30} - \frac{271525895659264597164129}{20449419753638432679768691} a^{26} - \frac{390044754339285825249518}{20449419753638432679768691} a^{22} + \frac{160260204316967552982876}{889105206679931855642117} a^{18} + \frac{352251185726095770867734}{889105206679931855642117} a^{14} + \frac{37292798999275391062173}{889105206679931855642117} a^{10} - \frac{269364708807597326628552}{889105206679931855642117} a^{6} - \frac{1747256001144836141445}{889105206679931855642117} a^{2}$, $\frac{1}{20449419753638432679768691} a^{43} - \frac{377495412904320220827686}{20449419753638432679768691} a^{39} - \frac{61573360947580349157897}{20449419753638432679768691} a^{35} + \frac{313099305151942484946038}{20449419753638432679768691} a^{31} - \frac{271525895659264597164129}{20449419753638432679768691} a^{27} - \frac{390044754339285825249518}{20449419753638432679768691} a^{23} + \frac{160260204316967552982876}{889105206679931855642117} a^{19} + \frac{352251185726095770867734}{889105206679931855642117} a^{15} + \frac{37292798999275391062173}{889105206679931855642117} a^{11} - \frac{269364708807597326628552}{889105206679931855642117} a^{7} - \frac{1747256001144836141445}{889105206679931855642117} a^{3}$  Toggle raw display

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:  \( \frac{2159754831367634313897}{20449419753638432679768691} a^{42} + \frac{149047263848576640299926}{20449419753638432679768691} a^{38} + \frac{4273665809431876600479477}{20449419753638432679768691} a^{34} + \frac{2868081116953286494019238}{889105206679931855642117} a^{30} + \frac{594053187211700567170909452}{20449419753638432679768691} a^{26} + \frac{3166997370442965220709483272}{20449419753638432679768691} a^{22} + \frac{422408163695724435246179921}{889105206679931855642117} a^{18} + \frac{694100615457853045962198762}{889105206679931855642117} a^{14} + \frac{536315973527527086028235889}{889105206679931855642117} a^{10} + \frac{156054186041719117834635830}{889105206679931855642117} a^{6} + \frac{12830402239602083026931771}{889105206679931855642117} a^{2} \) (order $4$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed  Toggle raw display
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{46}) \), \(\Q(i, \sqrt{46})\), \(\Q(\zeta_{23})^+\), 22.0.7198079267989980836471065337135104.1, 22.0.339058325839400057321133061640411938816.1, 22.22.339058325839400057321133061640411938816.1

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 $22^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{4}$ $22^{2}$

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

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