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Properties

Label 16.0.35087248320...6729.2
Degree $16$
Signature $[0, 8]$
Discriminant $67^{4}\cdot 89^{15}$
Root discriminant $192.34$
Ramified primes $67, 89$
Class number $113$ (GRH)
Class group $[113]$ (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![7902260224, 170454528, 1308458752, -241209680, 247420336, 16905736, 14729992, -265833, -626929, 456417, 21231, -9042, 754, 169, -41, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 41*x^14 + 169*x^13 + 754*x^12 - 9042*x^11 + 21231*x^10 + 456417*x^9 - 626929*x^8 - 265833*x^7 + 14729992*x^6 + 16905736*x^5 + 247420336*x^4 - 241209680*x^3 + 1308458752*x^2 + 170454528*x + 7902260224)
 
gp: K = bnfinit(x^16 - 7*x^15 - 41*x^14 + 169*x^13 + 754*x^12 - 9042*x^11 + 21231*x^10 + 456417*x^9 - 626929*x^8 - 265833*x^7 + 14729992*x^6 + 16905736*x^5 + 247420336*x^4 - 241209680*x^3 + 1308458752*x^2 + 170454528*x + 7902260224, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 41 x^{14} + 169 x^{13} + 754 x^{12} - 9042 x^{11} + 21231 x^{10} + 456417 x^{9} - 626929 x^{8} - 265833 x^{7} + 14729992 x^{6} + 16905736 x^{5} + 247420336 x^{4} - 241209680 x^{3} + 1308458752 x^{2} + 170454528 x + 7902260224 \)

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:  \(3508724832059361738820170787833596729=67^{4}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $192.34$
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:  $67, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{3}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{16} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{3}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{32} a^{8} - \frac{1}{32} a^{7} - \frac{15}{64} a^{5} - \frac{1}{64} a^{4} - \frac{7}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{10} - \frac{1}{32} a^{9} - \frac{1}{32} a^{7} - \frac{3}{64} a^{6} + \frac{1}{16} a^{5} + \frac{15}{64} a^{4} - \frac{3}{16} a^{2}$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{11} - \frac{1}{64} a^{10} - \frac{1}{32} a^{9} - \frac{1}{64} a^{8} + \frac{1}{128} a^{7} + \frac{31}{128} a^{5} - \frac{3}{32} a^{4} + \frac{9}{32} a^{3} - \frac{3}{8} a^{2}$, $\frac{1}{512} a^{14} - \frac{1}{256} a^{13} - \frac{1}{512} a^{12} + \frac{7}{256} a^{9} - \frac{11}{512} a^{8} + \frac{3}{256} a^{7} - \frac{25}{512} a^{6} - \frac{29}{256} a^{5} - \frac{23}{128} a^{4} - \frac{3}{64} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{3129120134505965419762886289855359983516672290829208808448} a^{15} + \frac{576346024688689535122864704602390828242699475776003489}{1564560067252982709881443144927679991758336145414604404224} a^{14} + \frac{491441615825310745561823697801244728745590176973132165}{284465466773269583614807844532305453046970208257200800768} a^{13} - \frac{5950056473348972237222827865189286589809707112562732107}{782280033626491354940721572463839995879168072707302202112} a^{12} + \frac{1977794570991989581340865546881577075476306173107144149}{391140016813245677470360786231919997939584036353651101056} a^{11} + \frac{38774172609827427246296668871264675023675536902707506503}{1564560067252982709881443144927679991758336145414604404224} a^{10} + \frac{54331264585162382062009028387137320636717393220427858141}{3129120134505965419762886289855359983516672290829208808448} a^{9} - \frac{28591451680669538173401755213677686990714218223969967243}{1564560067252982709881443144927679991758336145414604404224} a^{8} + \frac{87307061586682678862710702334511425840607523073944313567}{3129120134505965419762886289855359983516672290829208808448} a^{7} - \frac{71794491779518736528488090384553645215285776784725248691}{1564560067252982709881443144927679991758336145414604404224} a^{6} - \frac{188493696957401721346492825179752088153645156216452669415}{782280033626491354940721572463839995879168072707302202112} a^{5} - \frac{12117513093572487290184127692515493797088007000985978301}{391140016813245677470360786231919997939584036353651101056} a^{4} - \frac{8950173248508014695729085180046936036584102711844711549}{48892502101655709683795098278989999742448004544206387632} a^{3} - \frac{4310191451088188120544257319662906172073426369567839931}{12223125525413927420948774569747499935612001136051596908} a^{2} - \frac{4508872528061426075545114804438487239544934577014025735}{12223125525413927420948774569747499935612001136051596908} a + \frac{201108502336868363637044521482114535535766303581193157}{3055781381353481855237193642436874983903000284012899227}$

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

Class group and class number

$C_{113}$, which has order $113$ (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:  \( 1761562788020 \) (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

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.0.44231334895529.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
67Data not computed
89Data not computed