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Properties

Label 44.0.911...456.1
Degree $44$
Signature $[0, 22]$
Discriminant $9.115\times 10^{80}$
Root discriminant $69.18$
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 + 61*x^40 + 1522*x^36 + 20041*x^32 + 150032*x^28 + 642172*x^24 + 1506232*x^20 + 1760035*x^16 + 860639*x^12 + 127699*x^8 + 2926*x^4 + 1)
 
gp: K = bnfinit(x^44 + 61*x^40 + 1522*x^36 + 20041*x^32 + 150032*x^28 + 642172*x^24 + 1506232*x^20 + 1760035*x^16 + 860639*x^12 + 127699*x^8 + 2926*x^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 2926, 0, 0, 0, 127699, 0, 0, 0, 860639, 0, 0, 0, 1760035, 0, 0, 0, 1506232, 0, 0, 0, 642172, 0, 0, 0, 150032, 0, 0, 0, 20041, 0, 0, 0, 1522, 0, 0, 0, 61, 0, 0, 0, 1]);
 

\(x^{44} + 61 x^{40} + 1522 x^{36} + 20041 x^{32} + 150032 x^{28} + 642172 x^{24} + 1506232 x^{20} + 1760035 x^{16} + 860639 x^{12} + 127699 x^{8} + 2926 x^{4} + 1\)  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:  \(911\!\cdots\!456\)\(\medspace = 2^{88}\cdot 23^{40}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $69.18$
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$$\chi_{184}(1,·)$, $\chi_{184}(3,·)$, $\chi_{184}(133,·)$, $\chi_{184}(9,·)$, $\chi_{184}(139,·)$, $\chi_{184}(13,·)$, $\chi_{184}(131,·)$, $\chi_{184}(151,·)$, $\chi_{184}(25,·)$, $\chi_{184}(27,·)$, $\chi_{184}(29,·)$, $\chi_{184}(31,·)$, $\chi_{184}(35,·)$, $\chi_{184}(165,·)$, $\chi_{184}(39,·)$, $\chi_{184}(41,·)$, $\chi_{184}(173,·)$, $\chi_{184}(47,·)$, $\chi_{184}(49,·)$, $\chi_{184}(179,·)$, $\chi_{184}(55,·)$, $\chi_{184}(59,·)$, $\chi_{184}(71,·)$, $\chi_{184}(73,·)$, $\chi_{184}(75,·)$, $\chi_{184}(77,·)$, $\chi_{184}(141,·)$, $\chi_{184}(81,·)$, $\chi_{184}(163,·)$, $\chi_{184}(85,·)$, $\chi_{184}(87,·)$, $\chi_{184}(169,·)$, $\chi_{184}(93,·)$, $\chi_{184}(95,·)$, $\chi_{184}(101,·)$, $\chi_{184}(105,·)$, $\chi_{184}(167,·)$, $\chi_{184}(177,·)$, $\chi_{184}(147,·)$, $\chi_{184}(117,·)$, $\chi_{184}(119,·)$, $\chi_{184}(121,·)$, $\chi_{184}(123,·)$, $\chi_{184}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{3844747107219467355553841461} a^{40} - \frac{329310562655152745086055257}{3844747107219467355553841461} a^{36} - \frac{907271928316941277326394910}{3844747107219467355553841461} a^{32} - \frac{1259912066745346167321069242}{3844747107219467355553841461} a^{28} + \frac{115350598868941151020844659}{3844747107219467355553841461} a^{24} + \frac{1169497223927233062531165844}{3844747107219467355553841461} a^{20} - \frac{290264640292738122201123587}{3844747107219467355553841461} a^{16} + \frac{1368291019434541582126756795}{3844747107219467355553841461} a^{12} + \frac{467482415138977146564746469}{3844747107219467355553841461} a^{8} - \frac{467973473429468979839811678}{3844747107219467355553841461} a^{4} + \frac{1453838531030461051742161957}{3844747107219467355553841461}$, $\frac{1}{3844747107219467355553841461} a^{41} - \frac{329310562655152745086055257}{3844747107219467355553841461} a^{37} - \frac{907271928316941277326394910}{3844747107219467355553841461} a^{33} - \frac{1259912066745346167321069242}{3844747107219467355553841461} a^{29} + \frac{115350598868941151020844659}{3844747107219467355553841461} a^{25} + \frac{1169497223927233062531165844}{3844747107219467355553841461} a^{21} - \frac{290264640292738122201123587}{3844747107219467355553841461} a^{17} + \frac{1368291019434541582126756795}{3844747107219467355553841461} a^{13} + \frac{467482415138977146564746469}{3844747107219467355553841461} a^{9} - \frac{467973473429468979839811678}{3844747107219467355553841461} a^{5} + \frac{1453838531030461051742161957}{3844747107219467355553841461} a$, $\frac{1}{3844747107219467355553841461} a^{42} - \frac{329310562655152745086055257}{3844747107219467355553841461} a^{38} - \frac{907271928316941277326394910}{3844747107219467355553841461} a^{34} - \frac{1259912066745346167321069242}{3844747107219467355553841461} a^{30} + \frac{115350598868941151020844659}{3844747107219467355553841461} a^{26} + \frac{1169497223927233062531165844}{3844747107219467355553841461} a^{22} - \frac{290264640292738122201123587}{3844747107219467355553841461} a^{18} + \frac{1368291019434541582126756795}{3844747107219467355553841461} a^{14} + \frac{467482415138977146564746469}{3844747107219467355553841461} a^{10} - \frac{467973473429468979839811678}{3844747107219467355553841461} a^{6} + \frac{1453838531030461051742161957}{3844747107219467355553841461} a^{2}$, $\frac{1}{3844747107219467355553841461} a^{43} - \frac{329310562655152745086055257}{3844747107219467355553841461} a^{39} - \frac{907271928316941277326394910}{3844747107219467355553841461} a^{35} - \frac{1259912066745346167321069242}{3844747107219467355553841461} a^{31} + \frac{115350598868941151020844659}{3844747107219467355553841461} a^{27} + \frac{1169497223927233062531165844}{3844747107219467355553841461} a^{23} - \frac{290264640292738122201123587}{3844747107219467355553841461} a^{19} + \frac{1368291019434541582126756795}{3844747107219467355553841461} a^{15} + \frac{467482415138977146564746469}{3844747107219467355553841461} a^{11} - \frac{467973473429468979839811678}{3844747107219467355553841461} a^{7} + \frac{1453838531030461051742161957}{3844747107219467355553841461} a^{3}$  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:  \( \frac{559364217950633152513003118}{3844747107219467355553841461} a^{43} + \frac{34120009129382592888858609622}{3844747107219467355553841461} a^{39} + \frac{851278791029073106538382179100}{3844747107219467355553841461} a^{35} + \frac{11208388430103503940586671208606}{3844747107219467355553841461} a^{31} + \frac{83898538381950787425688419125632}{3844747107219467355553841461} a^{27} + \frac{359029571555533259619041645454461}{3844747107219467355553841461} a^{23} + \frac{841776480195238370324038807360508}{3844747107219467355553841461} a^{19} + \frac{982761154071312182014857524570016}{3844747107219467355553841461} a^{15} + \frac{479453552318492415604964710844738}{3844747107219467355553841461} a^{11} + \frac{70553971991710607652936942463583}{3844747107219467355553841461} a^{7} + \frac{1536643884990641141797014151784}{3844747107219467355553841461} a^{3} \) (order $8$)  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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{8})\), \(\Q(\zeta_{23})^+\), 22.0.7198079267989980836471065337135104.1, 22.0.14741666340843480753092741810452692992.1, 22.22.14741666340843480753092741810452692992.1

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 $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{4}$ $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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
23Data not computed