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Properties

Label 16.0.26436931289...625.14
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{8}$
Root discriminant $33.60$
Ramified primes $5, 101$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6091, -2668, 3571, 4773, -415, 1307, 2523, -576, -82, 441, -62, -204, 44, 39, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 8*x^14 + 39*x^13 + 44*x^12 - 204*x^11 - 62*x^10 + 441*x^9 - 82*x^8 - 576*x^7 + 2523*x^6 + 1307*x^5 - 415*x^4 + 4773*x^3 + 3571*x^2 - 2668*x + 6091)
 
gp: K = bnfinit(x^16 - 4*x^15 - 8*x^14 + 39*x^13 + 44*x^12 - 204*x^11 - 62*x^10 + 441*x^9 - 82*x^8 - 576*x^7 + 2523*x^6 + 1307*x^5 - 415*x^4 + 4773*x^3 + 3571*x^2 - 2668*x + 6091, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 8 x^{14} + 39 x^{13} + 44 x^{12} - 204 x^{11} - 62 x^{10} + 441 x^{9} - 82 x^{8} - 576 x^{7} + 2523 x^{6} + 1307 x^{5} - 415 x^{4} + 4773 x^{3} + 3571 x^{2} - 2668 x + 6091 \)

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 a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{55} a^{13} - \frac{2}{55} a^{12} - \frac{24}{55} a^{11} + \frac{5}{11} a^{10} + \frac{21}{55} a^{9} + \frac{24}{55} a^{8} - \frac{16}{55} a^{7} + \frac{1}{55} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{5}{11} a^{3} + \frac{4}{55} a^{2} - \frac{3}{55} a + \frac{19}{55}$, $\frac{1}{3905} a^{14} - \frac{14}{3905} a^{13} + \frac{6}{71} a^{12} + \frac{1138}{3905} a^{11} - \frac{169}{3905} a^{10} + \frac{817}{3905} a^{9} + \frac{906}{3905} a^{8} - \frac{12}{55} a^{7} - \frac{111}{3905} a^{6} - \frac{158}{355} a^{5} - \frac{1933}{3905} a^{4} + \frac{1409}{3905} a^{3} - \frac{491}{3905} a^{2} - \frac{12}{71} a + \frac{212}{3905}$, $\frac{1}{579092795430151108037525} a^{15} - \frac{3369321513574538857}{52644799584559191639775} a^{14} - \frac{1525720653623487321067}{579092795430151108037525} a^{13} - \frac{601343173932057344444}{10528959916911838327955} a^{12} + \frac{128276947449881488721634}{579092795430151108037525} a^{11} - \frac{253770633819237238754486}{579092795430151108037525} a^{10} + \frac{116608593722411137884586}{579092795430151108037525} a^{9} + \frac{112723319474064815117108}{579092795430151108037525} a^{8} + \frac{113208492692003882188289}{579092795430151108037525} a^{7} - \frac{209281000666600432168143}{579092795430151108037525} a^{6} + \frac{22163492891202350582007}{579092795430151108037525} a^{5} + \frac{245945433292591363092541}{579092795430151108037525} a^{4} - \frac{157844804009299132218}{470424691657312029275} a^{3} - \frac{5229496845662751810183}{52644799584559191639775} a^{2} - \frac{29231284405251366073891}{115818559086030221607505} a - \frac{116819974845140305843933}{579092795430151108037525}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( 38120.6275869 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

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 $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 4.0.1275125.2, 4.0.12625.1, 8.8.16098453125.1, 8.0.643938125.1, 8.0.1625943765625.4

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$