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Properties

Label 16.4.10643773710...2521.1
Degree $16$
Signature $[4, 6]$
Discriminant $17^{14}\cdot 43^{6}$
Root discriminant $48.89$
Ramified primes $17, 43$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 340, 1601, -1564, 2222, 25262, 12775, -18904, -8865, -782, 2855, 680, 31, -102, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 - 102*x^13 + 31*x^12 + 680*x^11 + 2855*x^10 - 782*x^9 - 8865*x^8 - 18904*x^7 + 12775*x^6 + 25262*x^5 + 2222*x^4 - 1564*x^3 + 1601*x^2 + 340*x + 16)
 
gp: K = bnfinit(x^16 - 10*x^14 - 102*x^13 + 31*x^12 + 680*x^11 + 2855*x^10 - 782*x^9 - 8865*x^8 - 18904*x^7 + 12775*x^6 + 25262*x^5 + 2222*x^4 - 1564*x^3 + 1601*x^2 + 340*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 10 x^{14} - 102 x^{13} + 31 x^{12} + 680 x^{11} + 2855 x^{10} - 782 x^{9} - 8865 x^{8} - 18904 x^{7} + 12775 x^{6} + 25262 x^{5} + 2222 x^{4} - 1564 x^{3} + 1601 x^{2} + 340 x + 16 \)

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:  \(1064377371083527836156872521=17^{14}\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.89$
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:  $17, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2}$, $\frac{1}{40} a^{10} - \frac{1}{40} a^{9} - \frac{1}{40} a^{8} + \frac{1}{10} a^{7} - \frac{1}{20} a^{6} + \frac{1}{20} a^{5} - \frac{3}{40} a^{4} - \frac{17}{40} a^{3} + \frac{19}{40} a^{2} + \frac{1}{20} a + \frac{1}{5}$, $\frac{1}{160} a^{11} + \frac{1}{160} a^{10} + \frac{1}{80} a^{9} + \frac{1}{80} a^{8} + \frac{11}{160} a^{7} - \frac{17}{160} a^{6} + \frac{3}{80} a^{5} - \frac{9}{80} a^{4} - \frac{15}{32} a^{3} + \frac{9}{32} a^{2} - \frac{17}{40} a + \frac{1}{10}$, $\frac{1}{480} a^{12} - \frac{1}{480} a^{11} - \frac{1}{120} a^{10} + \frac{11}{240} a^{9} - \frac{29}{480} a^{8} + \frac{1}{96} a^{7} - \frac{13}{120} a^{6} + \frac{11}{240} a^{5} + \frac{11}{160} a^{4} - \frac{59}{160} a^{3} - \frac{67}{240} a^{2} + \frac{3}{10} a - \frac{7}{15}$, $\frac{1}{960} a^{13} - \frac{1}{480} a^{11} - \frac{1}{320} a^{10} - \frac{37}{960} a^{9} + \frac{1}{160} a^{8} - \frac{5}{96} a^{7} + \frac{9}{320} a^{6} + \frac{17}{192} a^{5} + \frac{29}{160} a^{4} - \frac{1}{120} a^{3} + \frac{229}{960} a^{2} + \frac{97}{240} a - \frac{23}{60}$, $\frac{1}{698880} a^{14} - \frac{103}{232960} a^{13} + \frac{61}{116480} a^{12} - \frac{163}{139776} a^{11} + \frac{19}{8320} a^{10} + \frac{39371}{698880} a^{9} - \frac{771}{11648} a^{8} - \frac{10873}{139776} a^{7} + \frac{1129}{34944} a^{6} + \frac{2999}{99840} a^{5} - \frac{5449}{26880} a^{4} - \frac{2323}{19968} a^{3} - \frac{7701}{46592} a^{2} + \frac{51817}{174720} a + \frac{16553}{43680}$, $\frac{1}{11652717672960} a^{15} - \frac{1533097}{2330543534592} a^{14} + \frac{594115219}{5826358836480} a^{13} - \frac{2274475327}{11652717672960} a^{12} - \frac{8598114463}{2913179418240} a^{11} - \frac{98577756533}{11652717672960} a^{10} + \frac{488823131}{83233697664} a^{9} + \frac{829655214851}{11652717672960} a^{8} - \frac{19673223089}{224090724480} a^{7} + \frac{483884043499}{3884239224320} a^{6} - \frac{470262342507}{1942119612160} a^{5} - \frac{1506159909673}{11652717672960} a^{4} + \frac{101495206587}{298787632640} a^{3} + \frac{1026178492771}{2913179418240} a^{2} - \frac{199328443021}{728294854560} a - \frac{13687518227}{91036856820}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 40031730.583 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.211259.1, 4.4.4913.1, 4.2.12427.1, 8.2.32624796874211.1, 8.2.32624796874211.2, 8.4.44630365081.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$