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Properties

Label 16.0.79100591003...0881.4
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![49455952218418176, 12600832442989824, 4598319446544384, 882336067964976, 168575519317340, 20641758474146, 2402287099291, 215209853765, 16277482567, 1429221904, 87537415, 6299195, 337161, 9272, 573, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 573*x^14 + 9272*x^13 + 337161*x^12 + 6299195*x^11 + 87537415*x^10 + 1429221904*x^9 + 16277482567*x^8 + 215209853765*x^7 + 2402287099291*x^6 + 20641758474146*x^5 + 168575519317340*x^4 + 882336067964976*x^3 + 4598319446544384*x^2 + 12600832442989824*x + 49455952218418176)
 
gp: K = bnfinit(x^16 - 3*x^15 + 573*x^14 + 9272*x^13 + 337161*x^12 + 6299195*x^11 + 87537415*x^10 + 1429221904*x^9 + 16277482567*x^8 + 215209853765*x^7 + 2402287099291*x^6 + 20641758474146*x^5 + 168575519317340*x^4 + 882336067964976*x^3 + 4598319446544384*x^2 + 12600832442989824*x + 49455952218418176, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 573 x^{14} + 9272 x^{13} + 337161 x^{12} + 6299195 x^{11} + 87537415 x^{10} + 1429221904 x^{9} + 16277482567 x^{8} + 215209853765 x^{7} + 2402287099291 x^{6} + 20641758474146 x^{5} + 168575519317340 x^{4} + 882336067964976 x^{3} + 4598319446544384 x^{2} + 12600832442989824 x + 49455952218418176 \)

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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{5} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{6} + \frac{1}{12} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{48} a^{9} - \frac{1}{48} a^{8} - \frac{1}{24} a^{7} + \frac{1}{48} a^{6} + \frac{1}{24} a^{5} - \frac{7}{48} a^{4} + \frac{1}{48} a^{3} - \frac{7}{24} a^{2} - \frac{5}{12} a$, $\frac{1}{96} a^{10} - \frac{1}{96} a^{9} - \frac{1}{48} a^{8} + \frac{1}{96} a^{7} + \frac{1}{48} a^{6} - \frac{7}{96} a^{5} + \frac{1}{96} a^{4} - \frac{7}{48} a^{3} + \frac{7}{24} a^{2}$, $\frac{1}{288} a^{11} - \frac{1}{288} a^{10} + \frac{1}{144} a^{9} - \frac{1}{96} a^{8} + \frac{5}{144} a^{7} - \frac{3}{32} a^{6} - \frac{55}{288} a^{5} - \frac{11}{48} a^{4} - \frac{1}{6} a^{3} - \frac{4}{9} a^{2} - \frac{1}{2} a$, $\frac{1}{1728} a^{12} - \frac{1}{864} a^{11} - \frac{1}{288} a^{10} - \frac{1}{216} a^{9} + \frac{43}{1728} a^{8} - \frac{23}{864} a^{7} + \frac{79}{864} a^{6} - \frac{59}{432} a^{5} + \frac{55}{576} a^{4} - \frac{211}{864} a^{3} - \frac{187}{432} a^{2} + \frac{5}{18} a$, $\frac{1}{672192} a^{13} - \frac{47}{336096} a^{12} + \frac{281}{336096} a^{11} - \frac{739}{336096} a^{10} - \frac{4339}{672192} a^{9} + \frac{1805}{112032} a^{8} + \frac{473}{84024} a^{7} - \frac{727}{56016} a^{6} - \frac{10889}{672192} a^{5} - \frac{29987}{168048} a^{4} - \frac{1807}{56016} a^{3} + \frac{19367}{42012} a^{2} - \frac{6683}{14004} a - \frac{161}{389}$, $\frac{1}{22182336} a^{14} - \frac{13}{22182336} a^{13} + \frac{2309}{11091168} a^{12} + \frac{2183}{11091168} a^{11} + \frac{4313}{22182336} a^{10} - \frac{34187}{7394112} a^{9} + \frac{36439}{1008288} a^{8} + \frac{17659}{924264} a^{7} - \frac{827231}{22182336} a^{6} + \frac{4298557}{22182336} a^{5} - \frac{618125}{3697056} a^{4} + \frac{1225487}{5545584} a^{3} + \frac{337}{21006} a^{2} + \frac{5711}{12837} a - \frac{1235}{4279}$, $\frac{1}{17571347910192745500080519664979996982270668421447853932868149313720619786367813057262830592} a^{15} + \frac{492277111668611224082508519154064161400871119635437728974957008442712495872997227}{72310073704496895062059751707736613095764067577974707542667281126422303647604168959929344} a^{14} - \frac{4095429990400174509013108311393732261296292034580984390040505679046126024538333712309}{5857115970064248500026839888326665660756889473815951310956049771240206595455937685754276864} a^{13} - \frac{217164178130218066354700920062994684372618484768383379113770031183562657242301276368849}{4392836977548186375020129916244999245567667105361963483217037328430154946591953264315707648} a^{12} - \frac{3644189678497124160483418133319219506830850037898056418919033010708671989502724591467533}{5857115970064248500026839888326665660756889473815951310956049771240206595455937685754276864} a^{11} - \frac{68383589855069627004343433980648554713353792277745307882173798932243958524564402672577553}{17571347910192745500080519664979996982270668421447853932868149313720619786367813057262830592} a^{10} - \frac{5459707664406561605586900913523544942618039732962563788556538964274626565974932597664797}{17571347910192745500080519664979996982270668421447853932868149313720619786367813057262830592} a^{9} + \frac{65838968532550103177863895971312958546297634608153741809105621329536969863993312461049759}{4392836977548186375020129916244999245567667105361963483217037328430154946591953264315707648} a^{8} + \frac{8816726470250536492179560014216268422933611314336278268763865186414708852463910471012727}{17571347910192745500080519664979996982270668421447853932868149313720619786367813057262830592} a^{7} + \frac{1232164617594372456898928364572368529946906071603238294129661749809269529660208043886759825}{17571347910192745500080519664979996982270668421447853932868149313720619786367813057262830592} a^{6} + \frac{3787969690819787119802175614952219901693681696560763876884703721014102309749917207563135135}{17571347910192745500080519664979996982270668421447853932868149313720619786367813057262830592} a^{5} + \frac{1503881509706982724772494104302260995229014337425902069993118534804811414737566039785893151}{8785673955096372750040259832489998491135334210723926966434074656860309893183906528631415296} a^{4} - \frac{1053974268406618745198862556576762820936969896139557253897728358799258533019148219634738251}{4392836977548186375020129916244999245567667105361963483217037328430154946591953264315707648} a^{3} - \frac{19156394327599125231456037341821992700545781337640762945242227021859997128901642807580893}{45758718516126941406459686627552075474663199014187119616844138837814114026999513169955288} a^{2} + \frac{1268205632722042704669393976683372375771473645137304390528152252107795102378321135870127}{2542151028673718967025538145975115304147955500788173312046896602100784112611084064997516} a + \frac{134138094682720578975694413953473143776671078792178608693569277285316440956302306710863}{282461225408190996336170905108346144905328388976463701338544066900087123623453784999724}$

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:  \( 27354159575.0 \) (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

\(\Q(\sqrt{4321}) \), 4.4.80677568161.1, 4.4.2781985109.1, 4.4.541460189.1, 8.0.28124827288894872783620641.1 x2, 8.8.6508870004372800921921.2

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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