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Properties

Label 16.0.76920193729...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 31^{12}$
Root discriminant $35.92$
Ramified primes $5, 31$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31769, -64942, 8379, 66066, -24511, -32172, 14333, 7692, -3716, -940, 395, 16, 55, 6, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 10*x^14 + 6*x^13 + 55*x^12 + 16*x^11 + 395*x^10 - 940*x^9 - 3716*x^8 + 7692*x^7 + 14333*x^6 - 32172*x^5 - 24511*x^4 + 66066*x^3 + 8379*x^2 - 64942*x + 31769)
 
gp: K = bnfinit(x^16 - 2*x^15 - 10*x^14 + 6*x^13 + 55*x^12 + 16*x^11 + 395*x^10 - 940*x^9 - 3716*x^8 + 7692*x^7 + 14333*x^6 - 32172*x^5 - 24511*x^4 + 66066*x^3 + 8379*x^2 - 64942*x + 31769, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 10 x^{14} + 6 x^{13} + 55 x^{12} + 16 x^{11} + 395 x^{10} - 940 x^{9} - 3716 x^{8} + 7692 x^{7} + 14333 x^{6} - 32172 x^{5} - 24511 x^{4} + 66066 x^{3} + 8379 x^{2} - 64942 x + 31769 \)

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:  \(7692019372935056259765625=5^{10}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.92$
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:  $5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a$, $\frac{1}{328} a^{14} - \frac{3}{328} a^{13} - \frac{7}{82} a^{12} - \frac{67}{328} a^{11} + \frac{27}{164} a^{10} + \frac{57}{328} a^{9} + \frac{3}{41} a^{8} - \frac{29}{328} a^{7} + \frac{73}{328} a^{6} - \frac{17}{41} a^{5} + \frac{18}{41} a^{4} + \frac{15}{82} a^{3} + \frac{121}{328} a^{2} - \frac{95}{328} a + \frac{29}{328}$, $\frac{1}{392112639870721903021605819464} a^{15} + \frac{102728357254353684956693453}{392112639870721903021605819464} a^{14} + \frac{5944386205870114175028607589}{196056319935360951510802909732} a^{13} + \frac{33313603540704592125831011739}{392112639870721903021605819464} a^{12} - \frac{347787737353033456373024133}{49014079983840237877700727433} a^{11} + \frac{8030144364471564282468573351}{392112639870721903021605819464} a^{10} + \frac{531712096379012267773432257}{3322988473480694093403439148} a^{9} + \frac{20784562557936362124123988233}{392112639870721903021605819464} a^{8} + \frac{35646850107382585079134457331}{392112639870721903021605819464} a^{7} + \frac{14152221632601958422289752411}{196056319935360951510802909732} a^{6} - \frac{46514517402144310566211151641}{98028159967680475755401454866} a^{5} + \frac{9707282364426415883893588858}{49014079983840237877700727433} a^{4} - \frac{88879225988102265658018933851}{392112639870721903021605819464} a^{3} - \frac{85468037204450674423568206807}{392112639870721903021605819464} a^{2} - \frac{57203034541902262202623086537}{392112639870721903021605819464} a + \frac{25969299652168953635451106439}{196056319935360951510802909732}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 614911.087883 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

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 $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{-31}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-155}) \), 4.0.4805.1 x2, 4.2.775.1 x2, \(\Q(\sqrt{5}, \sqrt{-31})\), 8.0.110937960125.1, 8.0.2773449003125.1, 8.0.577200625.1

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 8 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$31$31.8.6.2$x^{8} + 713 x^{4} + 138384$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
31.8.6.2$x^{8} + 713 x^{4} + 138384$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$