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Properties

Label 16.4.87678952778...7713.3
Degree $16$
Signature $[4, 6]$
Discriminant $61^{4}\cdot 97^{15}$
Root discriminant $203.67$
Ramified primes $61, 97$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125218271, 666267339, -1302349077, -181534839, -116028954, -76994271, -28706604, 5701089, -2353771, 330544, 3799, -13362, 1919, -652, 80, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 80*x^14 - 652*x^13 + 1919*x^12 - 13362*x^11 + 3799*x^10 + 330544*x^9 - 2353771*x^8 + 5701089*x^7 - 28706604*x^6 - 76994271*x^5 - 116028954*x^4 - 181534839*x^3 - 1302349077*x^2 + 666267339*x + 125218271)
 
gp: K = bnfinit(x^16 - 3*x^15 + 80*x^14 - 652*x^13 + 1919*x^12 - 13362*x^11 + 3799*x^10 + 330544*x^9 - 2353771*x^8 + 5701089*x^7 - 28706604*x^6 - 76994271*x^5 - 116028954*x^4 - 181534839*x^3 - 1302349077*x^2 + 666267339*x + 125218271, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 80 x^{14} - 652 x^{13} + 1919 x^{12} - 13362 x^{11} + 3799 x^{10} + 330544 x^{9} - 2353771 x^{8} + 5701089 x^{7} - 28706604 x^{6} - 76994271 x^{5} - 116028954 x^{4} - 181534839 x^{3} - 1302349077 x^{2} + 666267339 x + 125218271 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8767895277848913089642817986417897713=61^{4}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.67$
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:  $61, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{103} a^{14} + \frac{32}{103} a^{13} - \frac{50}{103} a^{12} + \frac{34}{103} a^{11} - \frac{2}{103} a^{10} - \frac{3}{103} a^{9} + \frac{14}{103} a^{8} + \frac{34}{103} a^{7} - \frac{51}{103} a^{6} + \frac{44}{103} a^{5} - \frac{1}{103} a^{4} + \frac{50}{103} a^{3} + \frac{44}{103} a^{2} - \frac{1}{103} a - \frac{12}{103}$, $\frac{1}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{15} - \frac{497390202164498876839490018826993933965499526246062804295458742585729}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{14} - \frac{146658920516360151339240070650873073513936357722861108374943179784573693}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{13} + \frac{8981989562611772185711403643618505965153427122434250597053933029532673}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{12} + \frac{113745555738618569056377785420596467842296526230864953295665357699883774}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{11} - \frac{40614127731701064324524516010051915678686839286789912663843242498091490}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{10} - \frac{90929421801894661556843886718194087882264495610102900911066003514837916}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{9} + \frac{122576485823209064048418440131653214307680329616497114328226533464796348}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{8} - \frac{56767560014145628245819401795864270093628804188695348974486786587120806}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{7} + \frac{160121390644254631212350932749860631147147149607069900929041008424576835}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{6} + \frac{64315289645090204169853135295135137130604739085134282771220405024305090}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{5} - \frac{31962467356729908585587386570389626313515063928882506558689752595050814}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{4} - \frac{154283467820672133928180983232348439842841712599632691462552246143088548}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{3} + \frac{43700693193507267201129428847377809875905771379126726998029956871703983}{328085039267960226355409649219365190069800737628664862762323787085499847} a^{2} - \frac{62326876461490645851031134636542983422670106773076864119964882905156009}{328085039267960226355409649219365190069800737628664862762323787085499847} a + \frac{91721634797501794979881169983196900485243632703387956115368022565571891}{328085039267960226355409649219365190069800737628664862762323787085499847}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $9$
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:  \( 123773712241 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
61Data not computed
97Data not computed