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Properties

Label 16.2.39852375631...0991.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,19^{8}\cdot 31^{15}$
Root discriminant $109.03$
Ramified primes $19, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43011072, -27340032, -2448320, 4368256, 817744, -1414400, -352708, 268072, 113531, -23595, -16813, 1088, 1220, -16, -47, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 47*x^14 - 16*x^13 + 1220*x^12 + 1088*x^11 - 16813*x^10 - 23595*x^9 + 113531*x^8 + 268072*x^7 - 352708*x^6 - 1414400*x^5 + 817744*x^4 + 4368256*x^3 - 2448320*x^2 - 27340032*x - 43011072)
 
gp: K = bnfinit(x^16 - x^15 - 47*x^14 - 16*x^13 + 1220*x^12 + 1088*x^11 - 16813*x^10 - 23595*x^9 + 113531*x^8 + 268072*x^7 - 352708*x^6 - 1414400*x^5 + 817744*x^4 + 4368256*x^3 - 2448320*x^2 - 27340032*x - 43011072, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 47 x^{14} - 16 x^{13} + 1220 x^{12} + 1088 x^{11} - 16813 x^{10} - 23595 x^{9} + 113531 x^{8} + 268072 x^{7} - 352708 x^{6} - 1414400 x^{5} + 817744 x^{4} + 4368256 x^{3} - 2448320 x^{2} - 27340032 x - 43011072 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-398523756312075451372168518540991=-\,19^{8}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $109.03$
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:  $19, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{24} a^{2} - \frac{1}{12} a$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{10} + \frac{1}{48} a^{9} - \frac{1}{4} a^{7} + \frac{1}{16} a^{5} - \frac{1}{16} a^{4} + \frac{11}{48} a^{3} - \frac{1}{6} a^{2} + \frac{5}{12} a$, $\frac{1}{384} a^{12} - \frac{1}{384} a^{11} - \frac{7}{384} a^{10} + \frac{1}{48} a^{9} - \frac{3}{32} a^{8} - \frac{1}{8} a^{7} - \frac{31}{128} a^{6} + \frac{7}{128} a^{5} - \frac{13}{384} a^{4} - \frac{5}{24} a^{3} + \frac{1}{96} a^{2} - \frac{5}{24} a - \frac{1}{2}$, $\frac{1}{4608} a^{13} + \frac{1}{4608} a^{12} - \frac{41}{4608} a^{11} + \frac{13}{2304} a^{10} + \frac{19}{1152} a^{9} - \frac{5}{192} a^{8} + \frac{65}{1536} a^{7} - \frac{247}{1536} a^{6} - \frac{643}{4608} a^{5} - \frac{389}{2304} a^{4} - \frac{559}{1152} a^{3} + \frac{239}{576} a^{2} - \frac{59}{144} a - \frac{5}{12}$, $\frac{1}{7004160} a^{14} + \frac{199}{7004160} a^{13} - \frac{631}{1400832} a^{12} + \frac{5813}{1751040} a^{11} + \frac{2227}{175104} a^{10} - \frac{187}{48640} a^{9} - \frac{152431}{2334720} a^{8} - \frac{229297}{2334720} a^{7} - \frac{301949}{1400832} a^{6} - \frac{22151}{350208} a^{5} + \frac{149719}{875520} a^{4} + \frac{13997}{87552} a^{3} - \frac{4117}{437760} a^{2} + \frac{6463}{36480} a - \frac{141}{3040}$, $\frac{1}{37865117669213619962517087682560} a^{15} + \frac{481657978331498827472951}{12621705889737873320839029227520} a^{14} + \frac{1266694884588558897725251711}{37865117669213619962517087682560} a^{13} + \frac{6923648757816375435148666381}{18932558834606809981258543841280} a^{12} + \frac{1982959695863254294749789937}{262952206036205694184146442240} a^{11} + \frac{6601688194318934560127645051}{1183284927162925623828658990080} a^{10} + \frac{199414235361598647080584173623}{7573023533842723992503417536512} a^{9} - \frac{346017765324626815120782328051}{12621705889737873320839029227520} a^{8} + \frac{2926244893124299150529252523221}{37865117669213619962517087682560} a^{7} - \frac{63203308545429866954072285209}{420723529657929110694634307584} a^{6} + \frac{559139841544595996001363147497}{2366569854325851247657317980160} a^{5} - \frac{95536568611963110772442288431}{1183284927162925623828658990080} a^{4} + \frac{14710647169705139293402061491}{788856618108617082552439326720} a^{3} + \frac{523329910508823417491822871919}{1183284927162925623828658990080} a^{2} - \frac{27688310798879511543678759931}{59164246358146281191432949504} a - \frac{2519418739805766893510966681}{24651769315894283829763728960}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $8$
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:  \( 339885759220 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

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 $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{589}) \), 4.2.10754551.1, 8.2.3585471383559631.1

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 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $16$ R $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
31Data not computed