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Properties

Label 16.4.14106298732...0561.1
Degree $16$
Signature $[4, 6]$
Discriminant $3^{12}\cdot 61^{12}$
Root discriminant $49.76$
Ramified primes $3, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3285, -27567, -15192, 22626, 13008, -18261, -18375, -1287, 4030, 243, -892, -18, 111, -18, -19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 19*x^14 - 18*x^13 + 111*x^12 - 18*x^11 - 892*x^10 + 243*x^9 + 4030*x^8 - 1287*x^7 - 18375*x^6 - 18261*x^5 + 13008*x^4 + 22626*x^3 - 15192*x^2 - 27567*x - 3285)
 
gp: K = bnfinit(x^16 - 19*x^14 - 18*x^13 + 111*x^12 - 18*x^11 - 892*x^10 + 243*x^9 + 4030*x^8 - 1287*x^7 - 18375*x^6 - 18261*x^5 + 13008*x^4 + 22626*x^3 - 15192*x^2 - 27567*x - 3285, 1)
 

Normalized defining polynomial

\( x^{16} - 19 x^{14} - 18 x^{13} + 111 x^{12} - 18 x^{11} - 892 x^{10} + 243 x^{9} + 4030 x^{8} - 1287 x^{7} - 18375 x^{6} - 18261 x^{5} + 13008 x^{4} + 22626 x^{3} - 15192 x^{2} - 27567 x - 3285 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1410629873249683485564270561=3^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.76$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{21} a^{12} + \frac{2}{7} a^{11} + \frac{8}{21} a^{10} - \frac{2}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{21} a^{6} - \frac{1}{7} a^{5} + \frac{1}{21} a^{4} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{21} a^{13} - \frac{1}{3} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{5}{21} a^{7} + \frac{2}{7} a^{6} - \frac{2}{21} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{2}{7}$, $\frac{1}{1533} a^{14} - \frac{5}{219} a^{13} + \frac{8}{511} a^{12} - \frac{295}{1533} a^{11} - \frac{412}{1533} a^{10} - \frac{125}{511} a^{9} - \frac{760}{1533} a^{8} - \frac{538}{1533} a^{7} - \frac{197}{511} a^{6} - \frac{74}{219} a^{5} - \frac{512}{1533} a^{4} + \frac{65}{511} a^{3} - \frac{173}{511} a^{2} - \frac{88}{511} a + \frac{2}{7}$, $\frac{1}{132954691237467242699087238105} a^{15} + \frac{2314635096210173146864723}{14772743470829693633231915345} a^{14} - \frac{125499039243864765468211445}{26590938247493448539817447621} a^{13} + \frac{711921129802160394447633949}{44318230412489080899695746035} a^{12} + \frac{3670470962483636931740958617}{8863646082497816179939149207} a^{11} - \frac{17520406347900518798236748341}{44318230412489080899695746035} a^{10} - \frac{545742356707449922819349668}{132954691237467242699087238105} a^{9} - \frac{965393558794165821604144367}{14772743470829693633231915345} a^{8} + \frac{16306847573942010086753977984}{132954691237467242699087238105} a^{7} + \frac{8156710420269748271174508977}{44318230412489080899695746035} a^{6} + \frac{6949810278554563711134808444}{44318230412489080899695746035} a^{5} - \frac{14287172666302394581075330844}{44318230412489080899695746035} a^{4} - \frac{7779873180973412722073079352}{44318230412489080899695746035} a^{3} - \frac{215069778557313735989340082}{2110391924404241947604559335} a^{2} + \frac{7269950290657157852483705779}{14772743470829693633231915345} a + \frac{19365350689898762339336428}{40473269783095051049950453}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 50227843.565 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

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

\(\Q(\sqrt{61}) \), 4.4.2042829.1, 4.2.11163.1, 4.2.680943.1, 8.4.4173150323241.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$61$61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$