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Properties

Label 16.0.89161004482...0000.9
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 5^{8}$
Root discriminant $20.39$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $C_8:C_2^2$ (as 16T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![40, -80, -56, 408, -216, -1016, 2424, -2628, 1957, -1292, 760, -388, 186, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 186*x^12 - 388*x^11 + 760*x^10 - 1292*x^9 + 1957*x^8 - 2628*x^7 + 2424*x^6 - 1016*x^5 - 216*x^4 + 408*x^3 - 56*x^2 - 80*x + 40)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 186*x^12 - 388*x^11 + 760*x^10 - 1292*x^9 + 1957*x^8 - 2628*x^7 + 2424*x^6 - 1016*x^5 - 216*x^4 + 408*x^3 - 56*x^2 - 80*x + 40, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 186 x^{12} - 388 x^{11} + 760 x^{10} - 1292 x^{9} + 1957 x^{8} - 2628 x^{7} + 2424 x^{6} - 1016 x^{5} - 216 x^{4} + 408 x^{3} - 56 x^{2} - 80 x + 40 \)

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:  \(891610044825600000000=2^{32}\cdot 3^{12}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.39$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{28110940} a^{14} - \frac{7}{28110940} a^{13} + \frac{1924381}{28110940} a^{12} + \frac{501855}{5622188} a^{11} + \frac{150169}{2162380} a^{10} - \frac{2309321}{28110940} a^{9} + \frac{1790911}{28110940} a^{8} + \frac{6183653}{28110940} a^{7} - \frac{640916}{7027735} a^{6} + \frac{137041}{1081190} a^{5} - \frac{6498857}{14055470} a^{4} - \frac{580257}{14055470} a^{3} - \frac{291381}{7027735} a^{2} + \frac{10333}{127777} a + \frac{337022}{1405547}$, $\frac{1}{5537855180} a^{15} + \frac{7}{425988860} a^{14} + \frac{20062397}{1107571036} a^{13} - \frac{231240371}{2768927590} a^{12} - \frac{960909273}{5537855180} a^{11} + \frac{85664081}{553785518} a^{10} + \frac{1209135393}{5537855180} a^{9} - \frac{285137357}{2768927590} a^{8} - \frac{11291920}{276892759} a^{7} - \frac{1705281}{5537855180} a^{6} + \frac{83758557}{2768927590} a^{5} - \frac{568218714}{1384463795} a^{4} - \frac{422277134}{1384463795} a^{3} + \frac{414760282}{1384463795} a^{2} + \frac{579091}{1405547} a - \frac{107526544}{276892759}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{10834}{108119} a^{14} - \frac{75838}{108119} a^{13} + \frac{265103}{108119} a^{12} - \frac{604724}{108119} a^{11} + \frac{1269065}{108119} a^{10} - \frac{2609494}{108119} a^{9} + \frac{9875611}{216238} a^{8} - \frac{7647698}{108119} a^{7} + \frac{10874153}{108119} a^{6} - \frac{13416877}{108119} a^{5} + \frac{13690025}{216238} a^{4} + \frac{3285285}{108119} a^{3} - \frac{3204457}{108119} a^{2} + \frac{6530}{9829} a + \frac{948813}{108119} \) (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:  \( 15800.4377022 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), 4.0.2880.1, 4.0.320.1, \(\Q(i, \sqrt{15})\), 8.0.5971968000.1, 8.0.5971968000.2, 8.0.207360000.3

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.65$x^{8} + 4 x^{6} + 28 x^{4} + 20$$8$$1$$16$$QD_{16}$$[2, 2, 5/2]^{2}$
2.8.16.65$x^{8} + 4 x^{6} + 28 x^{4} + 20$$8$$1$$16$$QD_{16}$$[2, 2, 5/2]^{2}$
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$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}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$