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Properties

Label 16.0.18553202120...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{6}\cdot 41^{8}$
Root discriminant $50.61$
Ramified primes $5, 29, 41$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2.C_2^2:D_4$ (as 16T225)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16140569, -19330719, 19530533, -10644004, 5624423, -1825197, 658364, -92998, 31960, -345, 3377, -1011, 650, -160, 44, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 44*x^14 - 160*x^13 + 650*x^12 - 1011*x^11 + 3377*x^10 - 345*x^9 + 31960*x^8 - 92998*x^7 + 658364*x^6 - 1825197*x^5 + 5624423*x^4 - 10644004*x^3 + 19530533*x^2 - 19330719*x + 16140569)
 
gp: K = bnfinit(x^16 - 6*x^15 + 44*x^14 - 160*x^13 + 650*x^12 - 1011*x^11 + 3377*x^10 - 345*x^9 + 31960*x^8 - 92998*x^7 + 658364*x^6 - 1825197*x^5 + 5624423*x^4 - 10644004*x^3 + 19530533*x^2 - 19330719*x + 16140569, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 44 x^{14} - 160 x^{13} + 650 x^{12} - 1011 x^{11} + 3377 x^{10} - 345 x^{9} + 31960 x^{8} - 92998 x^{7} + 658364 x^{6} - 1825197 x^{5} + 5624423 x^{4} - 10644004 x^{3} + 19530533 x^{2} - 19330719 x + 16140569 \)

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:  \(1855320212000952785484765625=5^{8}\cdot 29^{6}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.61$
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:  $5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{41} a^{12} - \frac{11}{41} a^{11} - \frac{18}{41} a^{10} - \frac{8}{41} a^{9} - \frac{7}{41} a^{8} + \frac{4}{41} a^{7} + \frac{13}{41} a^{6} - \frac{3}{41} a^{5} + \frac{18}{41} a^{4} - \frac{16}{41} a^{3} - \frac{18}{41} a^{2} + \frac{5}{41} a + \frac{17}{41}$, $\frac{1}{451} a^{13} + \frac{4}{451} a^{12} - \frac{60}{451} a^{11} - \frac{114}{451} a^{10} + \frac{201}{451} a^{9} + \frac{63}{451} a^{8} + \frac{73}{451} a^{7} + \frac{28}{451} a^{6} + \frac{5}{41} a^{5} + \frac{8}{451} a^{4} + \frac{111}{451} a^{3} + \frac{145}{451} a^{2} + \frac{51}{451} a + \frac{50}{451}$, $\frac{1}{451} a^{14} + \frac{1}{451} a^{12} + \frac{181}{451} a^{11} + \frac{173}{451} a^{10} - \frac{4}{451} a^{9} + \frac{184}{451} a^{8} + \frac{4}{41} a^{7} + \frac{42}{451} a^{6} + \frac{8}{451} a^{5} + \frac{112}{451} a^{4} - \frac{178}{451} a^{3} - \frac{111}{451} a^{2} - \frac{20}{41} a + \frac{207}{451}$, $\frac{1}{298116485566642600124269861418152949615095177} a^{15} + \frac{40828505854736281193050670326945328513178}{298116485566642600124269861418152949615095177} a^{14} - \frac{79714088236251048604960105362017220723763}{298116485566642600124269861418152949615095177} a^{13} - \frac{919054104645077574319980425870223250309960}{298116485566642600124269861418152949615095177} a^{12} + \frac{818913767181066751738348556200999179766749}{7271133794308356100591947839467145112563297} a^{11} + \frac{80853456199372892823758188516644144000623230}{298116485566642600124269861418152949615095177} a^{10} - \frac{64314291479310362742776829825490746160814648}{298116485566642600124269861418152949615095177} a^{9} - \frac{126085759052507952324379228548971447645657447}{298116485566642600124269861418152949615095177} a^{8} + \frac{113429046034856804927690660736423054601277524}{298116485566642600124269861418152949615095177} a^{7} - \frac{82171476378301849103892468997801297216907}{1665455226629288268850669616861189662654163} a^{6} + \frac{4658322249018470206391484966566516323983799}{298116485566642600124269861418152949615095177} a^{5} + \frac{124162239544497275093436717099408773366051084}{298116485566642600124269861418152949615095177} a^{4} + \frac{5611115383975582275368932295292274941431464}{298116485566642600124269861418152949615095177} a^{3} + \frac{60188666593229235532224598757822568388975207}{298116485566642600124269861418152949615095177} a^{2} - \frac{82631340221979529928788068678848891897522741}{298116485566642600124269861418152949615095177} a - \frac{2920650087060732102481563264700523984086552}{7271133794308356100591947839467145112563297}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 2526845.66715 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^2:D_4$ (as 16T225):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^2.C_2^2:D_4$
Character table for $C_2^2.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.29725.2, 4.4.725.1, 4.0.1025.1, 8.0.883575625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$41$41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$