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Properties

Label 16.0.56411010787...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{4}\cdot 19^{8}$
Root discriminant $14.86$
Ramified primes $3, 5, 19$
Class number $1$
Class group Trivial
Galois group $C_2^2:D_4$ (as 16T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -2, -11, 58, -144, 187, -168, 196, -253, 220, -134, 83, -56, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 83*x^12 - 134*x^11 + 220*x^10 - 253*x^9 + 196*x^8 - 168*x^7 + 187*x^6 - 144*x^5 + 58*x^4 - 11*x^3 - 2*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 83*x^12 - 134*x^11 + 220*x^10 - 253*x^9 + 196*x^8 - 168*x^7 + 187*x^6 - 144*x^5 + 58*x^4 - 11*x^3 - 2*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 83 x^{12} - 134 x^{11} + 220 x^{10} - 253 x^{9} + 196 x^{8} - 168 x^{7} + 187 x^{6} - 144 x^{5} + 58 x^{4} - 11 x^{3} - 2 x^{2} + 3 x + 1 \)

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:  \(5641101078795050625=3^{12}\cdot 5^{4}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.86$
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:  $3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{14} a^{13} - \frac{3}{14} a^{12} - \frac{2}{7} a^{11} - \frac{1}{14} a^{10} - \frac{3}{14} a^{9} - \frac{2}{7} a^{8} + \frac{3}{14} a^{7} + \frac{3}{14} a^{6} - \frac{3}{7} a^{5} + \frac{5}{14} a^{4} + \frac{1}{14} a^{3} - \frac{2}{7} a^{2} + \frac{1}{14} a - \frac{5}{14}$, $\frac{1}{1946} a^{14} - \frac{1}{278} a^{13} + \frac{463}{1946} a^{12} - \frac{741}{1946} a^{11} + \frac{631}{1946} a^{10} - \frac{97}{1946} a^{9} + \frac{747}{1946} a^{8} - \frac{541}{1946} a^{7} + \frac{955}{1946} a^{6} - \frac{923}{1946} a^{5} + \frac{65}{1946} a^{4} + \frac{615}{1946} a^{3} + \frac{213}{1946} a^{2} + \frac{565}{1946} a - \frac{645}{1946}$, $\frac{1}{95354} a^{15} + \frac{17}{95354} a^{14} + \frac{1329}{47677} a^{13} + \frac{8452}{47677} a^{12} + \frac{20099}{95354} a^{11} - \frac{2986}{6811} a^{10} - \frac{443}{47677} a^{9} - \frac{38769}{95354} a^{8} - \frac{21930}{47677} a^{7} - \frac{3944}{47677} a^{6} - \frac{5129}{95354} a^{5} + \frac{21590}{47677} a^{4} + \frac{1857}{47677} a^{3} - \frac{15451}{95354} a^{2} + \frac{14450}{47677} a + \frac{27193}{95354}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{205}{973} a^{14} + \frac{205}{139} a^{13} - \frac{4426}{973} a^{12} + \frac{7901}{973} a^{11} - \frac{11622}{973} a^{10} + \frac{19885}{973} a^{9} - \frac{30537}{973} a^{8} + \frac{31119}{973} a^{7} - \frac{24527}{973} a^{6} + \frac{22832}{973} a^{5} - \frac{23055}{973} a^{4} + \frac{16956}{973} a^{3} - \frac{6691}{973} a^{2} + \frac{935}{973} a + \frac{870}{973} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1361.78824522 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T43):

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 $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{-3}) \), 4.0.1805.1, 4.0.16245.1, 4.0.513.1 x2, \(\Q(\sqrt{-3}, \sqrt{-19})\), 4.2.9747.1 x2, 8.0.95004009.1, 8.0.2375100225.1, 8.0.263900025.1

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 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$