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Properties

Label 16.0.94407327147...1296.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{14}\cdot 7^{6}$
Root discriminant $15.34$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.(C_2\times D_4)$ (as 16T408)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 12, 64, 125, 144, 172, 122, -30, -58, 22, 30, -4, -2, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 2*x^13 - 4*x^12 + 30*x^11 + 22*x^10 - 58*x^9 - 30*x^8 + 122*x^7 + 172*x^6 + 144*x^5 + 125*x^4 + 64*x^3 + 12*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 - 2*x^13 - 4*x^12 + 30*x^11 + 22*x^10 - 58*x^9 - 30*x^8 + 122*x^7 + 172*x^6 + 144*x^5 + 125*x^4 + 64*x^3 + 12*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} - 2 x^{13} - 4 x^{12} + 30 x^{11} + 22 x^{10} - 58 x^{9} - 30 x^{8} + 122 x^{7} + 172 x^{6} + 144 x^{5} + 125 x^{4} + 64 x^{3} + 12 x^{2} + 2 x + 1 \)

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:  \(9440732714731831296=2^{24}\cdot 3^{14}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.34$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{5}$, $\frac{1}{29872638747} a^{15} - \frac{496342332}{3319182083} a^{14} + \frac{954446212}{9957546249} a^{13} - \frac{2796232694}{29872638747} a^{12} + \frac{1109507479}{9957546249} a^{11} - \frac{1375162141}{9957546249} a^{10} + \frac{191461807}{29872638747} a^{9} - \frac{289373370}{3319182083} a^{8} - \frac{3258415043}{9957546249} a^{7} - \frac{13928720791}{29872638747} a^{6} - \frac{1478472106}{3319182083} a^{5} + \frac{230912237}{474168869} a^{4} + \frac{4009363616}{29872638747} a^{3} - \frac{4926106408}{9957546249} a^{2} + \frac{2730308597}{9957546249} a + \frac{10608413918}{29872638747}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{164599820}{33081549} a^{15} - \frac{754617458}{33081549} a^{14} + \frac{1426001926}{33081549} a^{13} - \frac{1150786594}{33081549} a^{12} - \frac{7199962}{33081549} a^{11} + \frac{4957986662}{33081549} a^{10} + \frac{723218330}{33081549} a^{9} - \frac{3348290475}{11027183} a^{8} + \frac{300067358}{11027183} a^{7} + \frac{19723696768}{33081549} a^{6} + \frac{16802443568}{33081549} a^{5} + \frac{13550161655}{33081549} a^{4} + \frac{12295284652}{33081549} a^{3} + \frac{3080662396}{33081549} a^{2} - \frac{25874282}{33081549} a + \frac{297477280}{33081549} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2379.04654143 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.(C_2\times D_4)$ (as 16T408):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^3.(C_2\times D_4)$
Character table for $C_2^3.(C_2\times D_4)$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.189.1, 4.0.1008.2, 4.0.432.1, 8.0.192036096.2, 8.0.192036096.1, 8.0.9144576.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$