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Properties

Label 16.0.52944586525...0001.6
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 89^{12}$
Root discriminant $96.10$
Ramified primes $11, 89$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1002752, 196096, 28144, -564240, 580060, -380752, 164141, -30720, 10054, -7222, 6445, -1932, -315, 266, -18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 18*x^14 + 266*x^13 - 315*x^12 - 1932*x^11 + 6445*x^10 - 7222*x^9 + 10054*x^8 - 30720*x^7 + 164141*x^6 - 380752*x^5 + 580060*x^4 - 564240*x^3 + 28144*x^2 + 196096*x + 1002752)
 
gp: K = bnfinit(x^16 - 8*x^15 - 18*x^14 + 266*x^13 - 315*x^12 - 1932*x^11 + 6445*x^10 - 7222*x^9 + 10054*x^8 - 30720*x^7 + 164141*x^6 - 380752*x^5 + 580060*x^4 - 564240*x^3 + 28144*x^2 + 196096*x + 1002752, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 18 x^{14} + 266 x^{13} - 315 x^{12} - 1932 x^{11} + 6445 x^{10} - 7222 x^{9} + 10054 x^{8} - 30720 x^{7} + 164141 x^{6} - 380752 x^{5} + 580060 x^{4} - 564240 x^{3} + 28144 x^{2} + 196096 x + 1002752 \)

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:  \(52944586525988002867127761920001=11^{8}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.10$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} + \frac{1}{16} a^{4} - \frac{7}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{3}{16} a^{5} + \frac{1}{8} a^{4} + \frac{1}{16} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{160} a^{12} + \frac{1}{40} a^{11} - \frac{1}{160} a^{10} - \frac{1}{16} a^{9} + \frac{1}{40} a^{8} - \frac{1}{80} a^{7} - \frac{11}{160} a^{6} + \frac{1}{20} a^{5} + \frac{23}{160} a^{4} - \frac{3}{10} a^{3} + \frac{3}{20} a^{2} + \frac{1}{20} a - \frac{2}{5}$, $\frac{1}{160} a^{13} + \frac{3}{160} a^{11} + \frac{1}{40} a^{10} - \frac{3}{80} a^{9} + \frac{1}{80} a^{8} - \frac{3}{160} a^{7} + \frac{3}{40} a^{6} + \frac{11}{160} a^{5} - \frac{3}{16} a^{4} - \frac{7}{80} a^{3} + \frac{3}{40} a^{2} - \frac{7}{20} a - \frac{2}{5}$, $\frac{1}{23084379881759360} a^{14} - \frac{7}{23084379881759360} a^{13} + \frac{67512995758907}{23084379881759360} a^{12} - \frac{405077974553351}{23084379881759360} a^{11} + \frac{217628524731481}{11542189940879680} a^{10} - \frac{674308982897923}{11542189940879680} a^{9} - \frac{69258077550041}{23084379881759360} a^{8} + \frac{205466979933749}{4616875976351872} a^{7} + \frac{2066877812730523}{23084379881759360} a^{6} - \frac{642733022638877}{4616875976351872} a^{5} - \frac{17905275273721}{721386871304980} a^{4} + \frac{1681784212848527}{5771094970439840} a^{3} - \frac{9939046932437}{721386871304980} a^{2} - \frac{54956027091557}{288554748521992} a + \frac{12067409170063}{360693435652490}$, $\frac{1}{6095822942236548917120} a^{15} + \frac{66013}{3047911471118274458560} a^{14} - \frac{17129029247261887}{47623616736223038415} a^{13} + \frac{128621336398759831}{190494466944892153660} a^{12} - \frac{25190864696694597009}{6095822942236548917120} a^{11} - \frac{4467751664084346281}{152395573555913722928} a^{10} - \frac{84342159808794181287}{6095822942236548917120} a^{9} + \frac{38601425220798413037}{761977867779568614640} a^{8} - \frac{5600055976542545323}{152395573555913722928} a^{7} + \frac{142233325634489393499}{3047911471118274458560} a^{6} + \frac{155659954239002295907}{6095822942236548917120} a^{5} - \frac{18202969294342416513}{95247233472446076830} a^{4} - \frac{368693062003742538097}{1523955735559137229280} a^{3} + \frac{120907107884672410027}{380988933889784307320} a^{2} + \frac{15445482687155893385}{76197786777956861464} a + \frac{14634888787838009009}{95247233472446076830}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 2158309583.94 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.2.87131.1, 4.0.85301249.2, 4.2.7754659.1, 8.4.81756214392809.1, 8.0.675671193329.1, 8.0.7276303080960001.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.1$x^{4} - 89$$4$$1$$3$$C_4$$[\ ]_{4}$