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Properties

Label 16.16.6688196296...7984.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{62}\cdot 449^{4}\cdot 1889^{2}$
Root discriminant $173.41$
Ramified primes $2, 449, 1889$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1162

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-87713567, 347099232, -289818592, -260086032, 359939588, 10037264, -101458480, 7284512, 11326242, -682112, -616768, 18000, 16476, -144, -208, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 208*x^14 - 144*x^13 + 16476*x^12 + 18000*x^11 - 616768*x^10 - 682112*x^9 + 11326242*x^8 + 7284512*x^7 - 101458480*x^6 + 10037264*x^5 + 359939588*x^4 - 260086032*x^3 - 289818592*x^2 + 347099232*x - 87713567)
 
gp: K = bnfinit(x^16 - 208*x^14 - 144*x^13 + 16476*x^12 + 18000*x^11 - 616768*x^10 - 682112*x^9 + 11326242*x^8 + 7284512*x^7 - 101458480*x^6 + 10037264*x^5 + 359939588*x^4 - 260086032*x^3 - 289818592*x^2 + 347099232*x - 87713567, 1)
 

Normalized defining polynomial

\( x^{16} - 208 x^{14} - 144 x^{13} + 16476 x^{12} + 18000 x^{11} - 616768 x^{10} - 682112 x^{9} + 11326242 x^{8} + 7284512 x^{7} - 101458480 x^{6} + 10037264 x^{5} + 359939588 x^{4} - 260086032 x^{3} - 289818592 x^{2} + 347099232 x - 87713567 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(668819629644740011880603666543017984=2^{62}\cdot 449^{4}\cdot 1889^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $173.41$
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:  $2, 449, 1889$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{8} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{8} - \frac{1}{2} a^{6} - \frac{3}{16} a^{4} - \frac{1}{16}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{9} - \frac{1}{2} a^{7} - \frac{3}{16} a^{5} - \frac{1}{16} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{10} - \frac{3}{16} a^{6} - \frac{1}{16} a^{2} - \frac{1}{2}$, $\frac{1}{104471876080662448215059607524672075132109763381744} a^{15} - \frac{1426986923389629990293194475161072626570166813879}{52235938040331224107529803762336037566054881690872} a^{14} - \frac{114683572207455528666728104437634704005783180167}{52235938040331224107529803762336037566054881690872} a^{13} + \frac{650190068756781377176698985943338023085926230769}{104471876080662448215059607524672075132109763381744} a^{12} - \frac{6417160779764781041711334910325372003236410001909}{104471876080662448215059607524672075132109763381744} a^{11} - \frac{1213201896416773147517892828370483419334647684395}{26117969020165612053764901881168018783027440845436} a^{10} + \frac{1739531654041664325732846702573679773467055295573}{52235938040331224107529803762336037566054881690872} a^{9} - \frac{2759962075241238028725375501951299018771931082021}{104471876080662448215059607524672075132109763381744} a^{8} - \frac{14874344310419360234945074447458703762722422111723}{104471876080662448215059607524672075132109763381744} a^{7} + \frac{16914182340212225672367736226511807445399176443955}{52235938040331224107529803762336037566054881690872} a^{6} - \frac{567863294276615646983284324900449259906269968895}{52235938040331224107529803762336037566054881690872} a^{5} + \frac{39606935995291337673965837574600051065028828917725}{104471876080662448215059607524672075132109763381744} a^{4} - \frac{44762495124698476133426480040949016173767815668589}{104471876080662448215059607524672075132109763381744} a^{3} - \frac{1974911978930634415954603942961318428340737865759}{13058984510082806026882450940584009391513720422718} a^{2} - \frac{10570270373083475316004050368323259587714047676459}{52235938040331224107529803762336037566054881690872} a - \frac{26489406902730393251333240502142777778786937874117}{104471876080662448215059607524672075132109763381744}$

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:  $15$
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:  \( 9934809769120 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1162:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1162
Character table for t16n1162 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.432934850920448.2

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.8$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.8$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
449Data not computed
1889Data not computed