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Properties

Label 16.0.79100591003...0881.3
Degree $16$
Signature $[0, 8]$
Discriminant $29^{14}\cdot 149^{14}$
Root discriminant $1517.53$
Ramified primes $29, 149$
Class number $11734105680$ (GRH)
Class group $[2, 24222, 242220]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31975694368889344, -21116314559719616, 7383199674276536, -1457129277599064, 281733439507432, -40295189035117, 5232393557019, -533426130836, 47662575288, -3445251317, 220735521, -11045491, 504403, -14970, 536, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 536*x^14 - 14970*x^13 + 504403*x^12 - 11045491*x^11 + 220735521*x^10 - 3445251317*x^9 + 47662575288*x^8 - 533426130836*x^7 + 5232393557019*x^6 - 40295189035117*x^5 + 281733439507432*x^4 - 1457129277599064*x^3 + 7383199674276536*x^2 - 21116314559719616*x + 31975694368889344)
 
gp: K = bnfinit(x^16 - 5*x^15 + 536*x^14 - 14970*x^13 + 504403*x^12 - 11045491*x^11 + 220735521*x^10 - 3445251317*x^9 + 47662575288*x^8 - 533426130836*x^7 + 5232393557019*x^6 - 40295189035117*x^5 + 281733439507432*x^4 - 1457129277599064*x^3 + 7383199674276536*x^2 - 21116314559719616*x + 31975694368889344, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 536 x^{14} - 14970 x^{13} + 504403 x^{12} - 11045491 x^{11} + 220735521 x^{10} - 3445251317 x^{9} + 47662575288 x^{8} - 533426130836 x^{7} + 5232393557019 x^{6} - 40295189035117 x^{5} + 281733439507432 x^{4} - 1457129277599064 x^{3} + 7383199674276536 x^{2} - 21116314559719616 x + 31975694368889344 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(791005910030165719912925444172568170567213001250881=29^{14}\cdot 149^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1517.53$
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:  $29, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{8} + \frac{1}{4} a^{2} - \frac{1}{3}$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{12} a^{2} + \frac{1}{6} a$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{1}{12} a^{7} + \frac{1}{6} a^{6} + \frac{1}{24} a^{5} - \frac{1}{12} a^{4} - \frac{1}{8} a^{3} + \frac{5}{12} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{96} a^{12} + \frac{1}{96} a^{10} - \frac{1}{48} a^{9} + \frac{1}{48} a^{8} - \frac{1}{12} a^{7} - \frac{11}{96} a^{6} - \frac{7}{48} a^{5} - \frac{3}{32} a^{4} + \frac{1}{6} a^{3} + \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{192} a^{13} - \frac{1}{192} a^{12} + \frac{1}{192} a^{11} - \frac{1}{64} a^{10} + \frac{1}{48} a^{9} + \frac{1}{32} a^{8} - \frac{1}{64} a^{7} - \frac{1}{64} a^{6} + \frac{5}{192} a^{5} + \frac{25}{192} a^{4} + \frac{5}{12} a^{3} + \frac{3}{8} a^{2} + \frac{1}{24} a$, $\frac{1}{2304} a^{14} - \frac{1}{2304} a^{13} - \frac{1}{768} a^{12} + \frac{37}{2304} a^{11} + \frac{1}{288} a^{10} - \frac{1}{128} a^{9} - \frac{11}{2304} a^{8} - \frac{115}{2304} a^{7} - \frac{575}{2304} a^{6} - \frac{125}{768} a^{5} + \frac{71}{576} a^{4} - \frac{73}{288} a^{3} + \frac{23}{96} a^{2} - \frac{23}{72} a + \frac{1}{9}$, $\frac{1}{3941360499211543025001594682553706308957957797985612610828434741361542158164503286589969067961878717952} a^{15} + \frac{7455098601326670187651439988011556977560479824226372059144904000664394009516941558754417659163653}{328446708267628585416799556879475525746496483165467717569036228446795179847041940549164088996823226496} a^{14} + \frac{890749417271854911965859593089108163352306662725441866117404362053829858196125651999959275663724083}{985340124802885756250398670638426577239489449496403152707108685340385539541125821647492266990469679488} a^{13} - \frac{9046284811130621061315573432722183577494038794726071912560005818442016647278497856214907559704488895}{1970680249605771512500797341276853154478978898992806305414217370680771079082251643294984533980939358976} a^{12} - \frac{23913624457676062727128813290583514416654894453585096033590220889232890516083617720128446984762554569}{1313786833070514341667198227517902102985985932661870870276144913787180719388167762196656355987292905984} a^{11} - \frac{29320015965820817819061227819942704917426939924012278931389499603596665433432766766184090344187548295}{1970680249605771512500797341276853154478978898992806305414217370680771079082251643294984533980939358976} a^{10} - \frac{90557827463230391527614207239317202757283290324946061835976094836931241850388182369871788497955297925}{3941360499211543025001594682553706308957957797985612610828434741361542158164503286589969067961878717952} a^{9} + \frac{5395485623240902978073618994549959414384089003032767999810883250909973703495580518023937074973227363}{218964472178419056944533037919650350497664322110311811712690818964530119898027960366109392664548817664} a^{8} + \frac{24179210038417923628008820506196537552258221213869307607545954370791843347789236145936612046750334921}{656893416535257170833599113758951051492992966330935435138072456893590359694083881098328177993646452992} a^{7} + \frac{200478641701063688812719349298327303725511095999833610230402411109604554239102160973774299666676616161}{1970680249605771512500797341276853154478978898992806305414217370680771079082251643294984533980939358976} a^{6} + \frac{382241881319960297686898987566226593441213574104947191088141001901483450704733207094365912653075888445}{3941360499211543025001594682553706308957957797985612610828434741361542158164503286589969067961878717952} a^{5} - \frac{6700132526901504019596311439611002836914558748864882429412678286618880510086271520183047260224180601}{82111677066907146354199889219868881436624120791366929392259057111698794961760485137291022249205806624} a^{4} + \frac{448889139443905677298897091267007032960134682792202774147042754044660868816637649351110024019133883}{1559082475953933158624048529491181293100457989709498659346690957817065727122034527923247257896312784} a^{3} - \frac{241908680490501999429563111032070931972592179625286007939703508593564512299455935990717201344629285367}{492670062401442878125199335319213288619744724748201576353554342670192769770562910823746133495234839744} a^{2} + \frac{4802544755073534652138114833848563453814274660230249890109864885907861258033017600841815653360339717}{10263959633363393294274986152483610179578015098920866174032382138962349370220060642161377781150725828} a + \frac{3453773317511604084365843002218742601551218020306599370053678565496594255129878747498788238398290953}{7697969725022544970706239614362707634683511324190649630524286604221762027665045481621033335863044371}$

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

Class group and class number

$C_{2}\times C_{24222}\times C_{242220}$, which has order $11734105680$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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:  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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 88555507897.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{4321}) \), 4.4.2781985109.1, 4.4.80677568161.2, 4.4.541460189.1, 8.0.28124827288894872783620641.2 x2, 8.8.6508870004372800921921.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
29Data not computed
149Data not computed