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Properties

Label 16.0.26436931289...625.20
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{8}$
Root discriminant $33.60$
Ramified primes $5, 101$
Class number $32$ (GRH)
Class group $[2, 2, 8]$ (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![119201, -27978, -14591, 52382, -25822, -5228, 16866, -7494, 1248, 132, -41, 26, 13, -16, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 16*x^13 + 13*x^12 + 26*x^11 - 41*x^10 + 132*x^9 + 1248*x^8 - 7494*x^7 + 16866*x^6 - 5228*x^5 - 25822*x^4 + 52382*x^3 - 14591*x^2 - 27978*x + 119201)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 - 16*x^13 + 13*x^12 + 26*x^11 - 41*x^10 + 132*x^9 + 1248*x^8 - 7494*x^7 + 16866*x^6 - 5228*x^5 - 25822*x^4 + 52382*x^3 - 14591*x^2 - 27978*x + 119201, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} - 16 x^{13} + 13 x^{12} + 26 x^{11} - 41 x^{10} + 132 x^{9} + 1248 x^{8} - 7494 x^{7} + 16866 x^{6} - 5228 x^{5} - 25822 x^{4} + 52382 x^{3} - 14591 x^{2} - 27978 x + 119201 \)

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:  \(2643693128974804931640625=5^{12}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.60$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{1}{10} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{9} - \frac{3}{10} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} - \frac{1}{10} a^{2} + \frac{3}{10} a + \frac{1}{5}$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{3}{10} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{3}{10}$, $\frac{1}{10} a^{11} - \frac{1}{2} a^{7} + \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{5} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{3}{10} a^{5} - \frac{3}{10} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{3550} a^{14} - \frac{33}{1775} a^{13} + \frac{13}{355} a^{12} - \frac{13}{3550} a^{11} + \frac{42}{1775} a^{10} - \frac{103}{3550} a^{9} - \frac{6}{1775} a^{8} + \frac{133}{355} a^{7} - \frac{113}{1775} a^{6} - \frac{651}{1775} a^{5} - \frac{707}{3550} a^{4} + \frac{1707}{3550} a^{3} + \frac{37}{142} a^{2} - \frac{329}{3550} a + \frac{427}{3550}$, $\frac{1}{1018404772461505702031709343550} a^{15} - \frac{38768628125909268524511053}{1018404772461505702031709343550} a^{14} + \frac{11292952930409546685364308806}{509202386230752851015854671775} a^{13} - \frac{22101257259128600968006507513}{1018404772461505702031709343550} a^{12} + \frac{2406788662915966679393225258}{101840477246150570203170934355} a^{11} + \frac{507351322265890569285123847}{509202386230752851015854671775} a^{10} + \frac{18465357310389878604871136997}{509202386230752851015854671775} a^{9} - \frac{2154329955588108870128790938}{509202386230752851015854671775} a^{8} + \frac{375006897871210625726569389509}{1018404772461505702031709343550} a^{7} + \frac{27845257706351033409976943093}{101840477246150570203170934355} a^{6} - \frac{266063106604493103386067633053}{1018404772461505702031709343550} a^{5} - \frac{177879513755882160113327761829}{1018404772461505702031709343550} a^{4} - \frac{210541691179501728244034331357}{509202386230752851015854671775} a^{3} + \frac{379202249250007069490216118111}{1018404772461505702031709343550} a^{2} - \frac{10047248309901550627066584171}{20368095449230114040634186871} a + \frac{25512071369307373446796149351}{1018404772461505702031709343550}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (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:  \( 8578.62068889 \) (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

\(\Q(\sqrt{5}) \), 4.4.2525.1, 4.0.1275125.2, 4.0.12625.1, 8.4.31878125.1, 8.4.325188753125.3, 8.0.1625943765625.4

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101Data not computed