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Properties

Label 16.0.11623374807...313.14
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 67^{8}$
Root discriminant $116.57$
Ramified primes $17, 67$
Class number $128$ (GRH)
Class group $[4, 32]$ (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![30880773, 14086248, -18353684, 1679760, 4768445, -2781018, 1075792, -314460, 59618, 2502, 1538, -2166, 667, -12, 2, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 2*x^14 - 12*x^13 + 667*x^12 - 2166*x^11 + 1538*x^10 + 2502*x^9 + 59618*x^8 - 314460*x^7 + 1075792*x^6 - 2781018*x^5 + 4768445*x^4 + 1679760*x^3 - 18353684*x^2 + 14086248*x + 30880773)
 
gp: K = bnfinit(x^16 - 6*x^15 + 2*x^14 - 12*x^13 + 667*x^12 - 2166*x^11 + 1538*x^10 + 2502*x^9 + 59618*x^8 - 314460*x^7 + 1075792*x^6 - 2781018*x^5 + 4768445*x^4 + 1679760*x^3 - 18353684*x^2 + 14086248*x + 30880773, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 2 x^{14} - 12 x^{13} + 667 x^{12} - 2166 x^{11} + 1538 x^{10} + 2502 x^{9} + 59618 x^{8} - 314460 x^{7} + 1075792 x^{6} - 2781018 x^{5} + 4768445 x^{4} + 1679760 x^{3} - 18353684 x^{2} + 14086248 x + 30880773 \)

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:  \(1162337480711184271576898233831313=17^{15}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.57$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{48} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{12} a + \frac{5}{16}$, $\frac{1}{240} a^{10} + \frac{1}{120} a^{9} + \frac{7}{80} a^{7} - \frac{1}{10} a^{6} - \frac{1}{4} a^{5} + \frac{1}{80} a^{4} - \frac{9}{40} a^{3} - \frac{5}{12} a^{2} + \frac{103}{240} a + \frac{9}{20}$, $\frac{1}{240} a^{11} + \frac{1}{240} a^{9} - \frac{3}{80} a^{8} - \frac{1}{40} a^{7} - \frac{9}{80} a^{6} - \frac{19}{80} a^{5} + \frac{23}{240} a^{3} + \frac{11}{80} a^{2} - \frac{29}{120} a + \frac{33}{80}$, $\frac{1}{480} a^{12} + \frac{1}{120} a^{9} + \frac{1}{20} a^{8} - \frac{1}{10} a^{7} - \frac{3}{80} a^{6} + \frac{1}{6} a^{4} + \frac{1}{40} a^{3} + \frac{3}{20} a^{2} - \frac{2}{15} a - \frac{21}{160}$, $\frac{1}{10560} a^{13} + \frac{1}{10560} a^{12} - \frac{1}{480} a^{11} + \frac{1}{880} a^{10} + \frac{17}{1760} a^{9} + \frac{169}{1760} a^{8} - \frac{41}{1760} a^{7} + \frac{51}{440} a^{6} - \frac{103}{480} a^{5} - \frac{211}{2640} a^{4} + \frac{1013}{5280} a^{3} - \frac{3}{160} a^{2} - \frac{1689}{3520} a + \frac{71}{320}$, $\frac{1}{54996480} a^{14} - \frac{17}{785664} a^{13} + \frac{271}{6110720} a^{12} - \frac{2699}{1833216} a^{11} + \frac{13}{2499840} a^{10} + \frac{61331}{6874560} a^{9} - \frac{32017}{327360} a^{8} - \frac{90653}{1309440} a^{7} - \frac{2067251}{27498240} a^{6} + \frac{424553}{5499648} a^{5} - \frac{470559}{3055360} a^{4} - \frac{72889}{2291520} a^{3} - \frac{4909417}{54996480} a^{2} + \frac{3231199}{6874560} a - \frac{34487}{238080}$, $\frac{1}{165534091322017079145191125017600} a^{15} + \frac{116677202973128457239687}{33106818264403415829038225003520} a^{14} - \frac{2229628271785472194148203621}{55178030440672359715063708339200} a^{13} + \frac{1667143094746686096178319227}{11035606088134471943012741667840} a^{12} - \frac{1835327234488241048630041247}{41383522830504269786297781254400} a^{11} + \frac{143923528899861791209701571283}{82767045661008539572595562508800} a^{10} - \frac{29458795960990233449448115207}{3448626902542022482191481771200} a^{9} + \frac{67308998040310312055472943423}{3941287888619454265361693452800} a^{8} - \frac{1828803059115525072910383136847}{20691761415252134893148890627200} a^{7} - \frac{91092813462099209675663742079}{41383522830504269786297781254400} a^{6} - \frac{294556131402322344758994819341}{1970643944309727132680846726400} a^{5} + \frac{395094647372053831045585260061}{9196338406778726619177284723200} a^{4} - \frac{3661087376703452667108804300707}{15048553756547007195017375001600} a^{3} + \frac{10440618484430786141199921589889}{23647727331716725592170160716800} a^{2} - \frac{17132893368061187703699030052547}{55178030440672359715063708339200} a + \frac{82092531007851137807394535579}{238865932643603288809799603200}$

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

Class group and class number

$C_{4}\times C_{32}$, which has order $128$ (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:  \( 5962737829.94 \) (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

\(\Q(\sqrt{-1139}) \), 4.0.22054457.1, 8.0.8268784250602433.4

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$67$67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$