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Properties

Label 44.44.1338316923...0000.1
Degree $44$
Signature $[44, 0]$
Discriminant $2^{66}\cdot 5^{33}\cdot 23^{42}$
Root discriminant $188.63$
Ramified primes $2, 5, 23$
Class number Not computed
Class group Not computed
Galois group $C_{44}$ (as 44T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![108339200000000000, 0, -1191731200000000000, 0, 4945684480000000000, 0, -10844753920000000000, 0, 14562955264000000000, 0, -13061238528000000000, 0, 8280581734400000000, 0, -3859279296000000000, 0, 1359724672000000000, 0, -369514540800000000, 0, 78572903232000000, 0, -13201383232000000, 0, 1762976128000000, 0, -187591348800000, 0, 15888506720000, 0, -1066130960000, 0, 56161768000, 0, -2288408000, 0, 70481200, 0, -1582400, 0, 24380, 0, -230, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 230*x^42 + 24380*x^40 - 1582400*x^38 + 70481200*x^36 - 2288408000*x^34 + 56161768000*x^32 - 1066130960000*x^30 + 15888506720000*x^28 - 187591348800000*x^26 + 1762976128000000*x^24 - 13201383232000000*x^22 + 78572903232000000*x^20 - 369514540800000000*x^18 + 1359724672000000000*x^16 - 3859279296000000000*x^14 + 8280581734400000000*x^12 - 13061238528000000000*x^10 + 14562955264000000000*x^8 - 10844753920000000000*x^6 + 4945684480000000000*x^4 - 1191731200000000000*x^2 + 108339200000000000)
 
gp: K = bnfinit(x^44 - 230*x^42 + 24380*x^40 - 1582400*x^38 + 70481200*x^36 - 2288408000*x^34 + 56161768000*x^32 - 1066130960000*x^30 + 15888506720000*x^28 - 187591348800000*x^26 + 1762976128000000*x^24 - 13201383232000000*x^22 + 78572903232000000*x^20 - 369514540800000000*x^18 + 1359724672000000000*x^16 - 3859279296000000000*x^14 + 8280581734400000000*x^12 - 13061238528000000000*x^10 + 14562955264000000000*x^8 - 10844753920000000000*x^6 + 4945684480000000000*x^4 - 1191731200000000000*x^2 + 108339200000000000, 1)
 

Normalized defining polynomial

\( x^{44} - 230 x^{42} + 24380 x^{40} - 1582400 x^{38} + 70481200 x^{36} - 2288408000 x^{34} + 56161768000 x^{32} - 1066130960000 x^{30} + 15888506720000 x^{28} - 187591348800000 x^{26} + 1762976128000000 x^{24} - 13201383232000000 x^{22} + 78572903232000000 x^{20} - 369514540800000000 x^{18} + 1359724672000000000 x^{16} - 3859279296000000000 x^{14} + 8280581734400000000 x^{12} - 13061238528000000000 x^{10} + 14562955264000000000 x^{8} - 10844753920000000000 x^{6} + 4945684480000000000 x^{4} - 1191731200000000000 x^{2} + 108339200000000000 \)

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

Invariants

Degree:  $44$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[44, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13383169230192059253459701104387771124501004765020501667165784506368000000000000000000000000000000000=2^{66}\cdot 5^{33}\cdot 23^{42}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $188.63$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(920=2^{3}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{920}(1,·)$, $\chi_{920}(773,·)$, $\chi_{920}(9,·)$, $\chi_{920}(209,·)$, $\chi_{920}(493,·)$, $\chi_{920}(917,·)$, $\chi_{920}(601,·)$, $\chi_{920}(409,·)$, $\chi_{920}(797,·)$, $\chi_{920}(517,·)$, $\chi_{920}(289,·)$, $\chi_{920}(37,·)$, $\chi_{920}(561,·)$, $\chi_{920}(809,·)$, $\chi_{920}(557,·)$, $\chi_{920}(157,·)$, $\chi_{920}(49,·)$, $\chi_{920}(53,·)$, $\chi_{920}(441,·)$, $\chi_{920}(573,·)$, $\chi_{920}(169,·)$, $\chi_{920}(373,·)$, $\chi_{920}(449,·)$, $\chi_{920}(841,·)$, $\chi_{920}(333,·)$, $\chi_{920}(721,·)$, $\chi_{920}(597,·)$, $\chi_{920}(121,·)$, $\chi_{920}(729,·)$, $\chi_{920}(361,·)$, $\chi_{920}(733,·)$, $\chi_{920}(677,·)$, $\chi_{920}(293,·)$, $\chi_{920}(613,·)$, $\chi_{920}(81,·)$, $\chi_{920}(489,·)$, $\chi_{920}(237,·)$, $\chi_{920}(369,·)$, $\chi_{920}(757,·)$, $\chi_{920}(41,·)$, $\chi_{920}(413,·)$, $\chi_{920}(761,·)$, $\chi_{920}(477,·)$, $\chi_{920}(893,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{20} a^{4}$, $\frac{1}{20} a^{5}$, $\frac{1}{40} a^{6}$, $\frac{1}{40} a^{7}$, $\frac{1}{400} a^{8}$, $\frac{1}{400} a^{9}$, $\frac{1}{800} a^{10}$, $\frac{1}{800} a^{11}$, $\frac{1}{8000} a^{12}$, $\frac{1}{8000} a^{13}$, $\frac{1}{16000} a^{14}$, $\frac{1}{16000} a^{15}$, $\frac{1}{160000} a^{16}$, $\frac{1}{160000} a^{17}$, $\frac{1}{320000} a^{18}$, $\frac{1}{320000} a^{19}$, $\frac{1}{3200000} a^{20}$, $\frac{1}{3200000} a^{21}$, $\frac{1}{147200000} a^{22}$, $\frac{1}{147200000} a^{23}$, $\frac{1}{1472000000} a^{24}$, $\frac{1}{1472000000} a^{25}$, $\frac{1}{2944000000} a^{26}$, $\frac{1}{2944000000} a^{27}$, $\frac{1}{29440000000} a^{28}$, $\frac{1}{29440000000} a^{29}$, $\frac{1}{58880000000} a^{30}$, $\frac{1}{58880000000} a^{31}$, $\frac{1}{588800000000} a^{32}$, $\frac{1}{588800000000} a^{33}$, $\frac{1}{1177600000000} a^{34}$, $\frac{1}{1177600000000} a^{35}$, $\frac{1}{11776000000000} a^{36}$, $\frac{1}{11776000000000} a^{37}$, $\frac{1}{23552000000000} a^{38}$, $\frac{1}{23552000000000} a^{39}$, $\frac{1}{32737280000000000} a^{40} - \frac{19}{1636864000000000} a^{38} + \frac{57}{1636864000000000} a^{36} - \frac{37}{163686400000000} a^{34} + \frac{7}{10230400000000} a^{32} + \frac{1}{1023040000000} a^{30} - \frac{33}{2046080000000} a^{28} + \frac{9}{204608000000} a^{26} - \frac{1}{3197000000} a^{24} + \frac{3}{5115200000} a^{22} - \frac{13}{88960000} a^{20} + \frac{61}{44480000} a^{18} - \frac{1}{5560000} a^{16} + \frac{61}{2224000} a^{14} + \frac{11}{1112000} a^{12} + \frac{7}{111200} a^{10} - \frac{7}{27800} a^{8} - \frac{27}{2780} a^{6} + \frac{3}{695} a^{4} - \frac{29}{278} a^{2} + \frac{50}{139}$, $\frac{1}{32737280000000000} a^{41} - \frac{19}{1636864000000000} a^{39} + \frac{57}{1636864000000000} a^{37} - \frac{37}{163686400000000} a^{35} + \frac{7}{10230400000000} a^{33} + \frac{1}{1023040000000} a^{31} - \frac{33}{2046080000000} a^{29} + \frac{9}{204608000000} a^{27} - \frac{1}{3197000000} a^{25} + \frac{3}{5115200000} a^{23} - \frac{13}{88960000} a^{21} + \frac{61}{44480000} a^{19} - \frac{1}{5560000} a^{17} + \frac{61}{2224000} a^{15} + \frac{11}{1112000} a^{13} + \frac{7}{111200} a^{11} - \frac{7}{27800} a^{9} - \frac{27}{2780} a^{7} + \frac{3}{695} a^{5} - \frac{29}{278} a^{3} + \frac{50}{139} a$, $\frac{1}{27987027405967360000000000} a^{42} - \frac{4101983}{3498378425745920000000000} a^{40} + \frac{18964067483}{1399351370298368000000000} a^{38} + \frac{2059442837}{349837842574592000000000} a^{36} - \frac{115769181}{349837842574592000000} a^{34} - \frac{300910469}{17491892128729600000000} a^{32} - \frac{5740952073}{1749189212872960000000} a^{30} + \frac{82769191}{38025852453760000000} a^{28} + \frac{9668509}{273310814511400000} a^{26} + \frac{17097125271}{87459460643648000000} a^{24} - \frac{13064830797}{8745946064364800000} a^{22} + \frac{17927745913}{190129262268800000} a^{20} - \frac{6433289177}{9506463113440000} a^{18} - \frac{200687567}{1188307889180000} a^{16} + \frac{11825438567}{475323155672000} a^{14} + \frac{609999133}{11883078891800} a^{12} + \frac{8842120783}{47532315567200} a^{10} - \frac{596365133}{594153944590} a^{8} + \frac{3923259251}{594153944590} a^{6} - \frac{22412302837}{1188307889180} a^{4} + \frac{7428755939}{118830788918} a^{2} + \frac{17364519196}{59415394459}$, $\frac{1}{27987027405967360000000000} a^{43} - \frac{4101983}{3498378425745920000000000} a^{41} + \frac{18964067483}{1399351370298368000000000} a^{39} + \frac{2059442837}{349837842574592000000000} a^{37} - \frac{115769181}{349837842574592000000} a^{35} - \frac{300910469}{17491892128729600000000} a^{33} - \frac{5740952073}{1749189212872960000000} a^{31} + \frac{82769191}{38025852453760000000} a^{29} + \frac{9668509}{273310814511400000} a^{27} + \frac{17097125271}{87459460643648000000} a^{25} - \frac{13064830797}{8745946064364800000} a^{23} + \frac{17927745913}{190129262268800000} a^{21} - \frac{6433289177}{9506463113440000} a^{19} - \frac{200687567}{1188307889180000} a^{17} + \frac{11825438567}{475323155672000} a^{15} + \frac{609999133}{11883078891800} a^{13} + \frac{8842120783}{47532315567200} a^{11} - \frac{596365133}{594153944590} a^{9} + \frac{3923259251}{594153944590} a^{7} - \frac{22412302837}{1188307889180} a^{5} + \frac{7428755939}{118830788918} a^{3} + \frac{17364519196}{59415394459} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $43$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{44}$ (as 44T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 44
The 44 conjugacy class representatives for $C_{44}$
Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4232000.1, \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1

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 $44$ R $44$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ $44$ $44$ $22^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $44$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ $44$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{11}$ $44$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{4}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed
23Data not computed