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Properties

Label 16.0.56192894500...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{6}\cdot 11^{8}$
Root discriminant $17.15$
Ramified primes $2, 5, 11$
Class number $1$
Class group Trivial
Galois group $C_2^2\wr C_2$ (as 16T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 4, -2, -9, 10, 68, 144, 197, 144, 68, 10, -9, -2, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 2*x^13 - 9*x^12 + 10*x^11 + 68*x^10 + 144*x^9 + 197*x^8 + 144*x^7 + 68*x^6 + 10*x^5 - 9*x^4 - 2*x^3 + 4*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 - 2*x^13 - 9*x^12 + 10*x^11 + 68*x^10 + 144*x^9 + 197*x^8 + 144*x^7 + 68*x^6 + 10*x^5 - 9*x^4 - 2*x^3 + 4*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} - 2 x^{13} - 9 x^{12} + 10 x^{11} + 68 x^{10} + 144 x^{9} + 197 x^{8} + 144 x^{7} + 68 x^{6} + 10 x^{5} - 9 x^{4} - 2 x^{3} + 4 x^{2} - 2 x + 1 \)

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:  \(56192894500864000000=2^{24}\cdot 5^{6}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.15$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{55} a^{12} + \frac{1}{55} a^{11} + \frac{1}{5} a^{10} + \frac{16}{55} a^{9} - \frac{14}{55} a^{8} - \frac{24}{55} a^{7} - \frac{1}{55} a^{6} + \frac{9}{55} a^{5} + \frac{19}{55} a^{4} - \frac{6}{55} a^{3} - \frac{1}{5} a^{2} - \frac{21}{55} a - \frac{2}{11}$, $\frac{1}{55} a^{13} - \frac{1}{55} a^{11} - \frac{6}{55} a^{10} + \frac{14}{55} a^{9} - \frac{21}{55} a^{8} + \frac{12}{55} a^{7} - \frac{1}{55} a^{6} + \frac{21}{55} a^{5} - \frac{14}{55} a^{4} + \frac{6}{55} a^{3} + \frac{1}{55} a^{2} + \frac{2}{5} a + \frac{21}{55}$, $\frac{1}{17545} a^{14} + \frac{93}{17545} a^{13} - \frac{94}{17545} a^{12} - \frac{1688}{17545} a^{11} + \frac{6562}{17545} a^{10} + \frac{8142}{17545} a^{9} + \frac{981}{3509} a^{8} - \frac{2659}{17545} a^{7} - \frac{1252}{3509} a^{6} - \frac{152}{17545} a^{5} - \frac{6517}{17545} a^{4} + \frac{7563}{17545} a^{3} + \frac{3734}{17545} a^{2} - \frac{7244}{17545} a - \frac{7336}{17545}$, $\frac{1}{87725} a^{15} - \frac{1}{87725} a^{14} - \frac{542}{87725} a^{13} - \frac{189}{87725} a^{12} - \frac{4793}{87725} a^{11} + \frac{1242}{87725} a^{10} - \frac{7773}{17545} a^{9} - \frac{22871}{87725} a^{8} - \frac{21084}{87725} a^{7} + \frac{1176}{3509} a^{6} + \frac{7133}{87725} a^{5} - \frac{24857}{87725} a^{4} + \frac{14709}{87725} a^{3} - \frac{27118}{87725} a^{2} + \frac{24116}{87725} a - \frac{31356}{87725}$

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:  \( \frac{87}{605} a^{15} - \frac{204}{605} a^{14} + \frac{386}{605} a^{13} - \frac{28}{55} a^{12} - \frac{130}{121} a^{11} + \frac{170}{121} a^{10} + \frac{6126}{605} a^{9} + \frac{10668}{605} a^{8} + \frac{10338}{605} a^{7} + \frac{76}{605} a^{6} - \frac{1623}{121} a^{5} - \frac{2000}{121} a^{4} - \frac{428}{55} a^{3} - \frac{824}{605} a^{2} + \frac{676}{605} a - \frac{133}{605} \) (order $4$)
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:  \( 3863.50483584 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T46):

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 $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-11}) \), 4.0.38720.3, 4.0.320.1, 4.4.38720.1, 4.0.38720.4, 4.0.605.1, \(\Q(i, \sqrt{11})\), 4.0.9680.1, 8.0.93702400.2, 8.0.1499238400.3, 8.0.1499238400.2

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 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$