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Properties

Label 16.0.94407327147...1296.5
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, -4, -6, -2, 53, 144, 208, 158, 78, -40, -44, -6, 8, -2, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 2*x^13 + 8*x^12 - 6*x^11 - 44*x^10 - 40*x^9 + 78*x^8 + 158*x^7 + 208*x^6 + 144*x^5 + 53*x^4 - 2*x^3 - 6*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 - 2*x^13 + 8*x^12 - 6*x^11 - 44*x^10 - 40*x^9 + 78*x^8 + 158*x^7 + 208*x^6 + 144*x^5 + 53*x^4 - 2*x^3 - 6*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} - 2 x^{13} + 8 x^{12} - 6 x^{11} - 44 x^{10} - 40 x^{9} + 78 x^{8} + 158 x^{7} + 208 x^{6} + 144 x^{5} + 53 x^{4} - 2 x^{3} - 6 x^{2} - 4 x + 1 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{5}$, $\frac{1}{60836968611} a^{15} + \frac{29708733}{6759663179} a^{14} + \frac{981926383}{6759663179} a^{13} + \frac{5204172769}{60836968611} a^{12} - \frac{519143297}{6759663179} a^{11} + \frac{691367579}{20278989537} a^{10} + \frac{5378588284}{60836968611} a^{9} + \frac{2710513847}{20278989537} a^{8} - \frac{176056334}{6759663179} a^{7} - \frac{11343191290}{60836968611} a^{6} - \frac{10095778639}{20278989537} a^{5} - \frac{2694823753}{6759663179} a^{4} + \frac{22348721246}{60836968611} a^{3} - \frac{5175658885}{20278989537} a^{2} - \frac{7464749588}{20278989537} a + \frac{3296404691}{60836968611}$

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:  \( -\frac{78689948}{33081549} a^{15} + \frac{292320554}{33081549} a^{14} - \frac{387430534}{33081549} a^{13} + \frac{41568362}{33081549} a^{12} - \frac{608848130}{33081549} a^{11} + \frac{291135412}{33081549} a^{10} + \frac{1188966396}{11027183} a^{9} + \frac{1382432197}{11027183} a^{8} - \frac{1668121498}{11027183} a^{7} - \frac{13928067292}{33081549} a^{6} - \frac{20277581600}{33081549} a^{5} - \frac{16878336185}{33081549} a^{4} - \frac{8597206394}{33081549} a^{3} - \frac{2030428390}{33081549} a^{2} + \frac{29642432}{33081549} a + \frac{116065936}{11027183} \) (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:  \( 3343.78532124 \) (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

\(\Q(\sqrt{-3}) \), 4.0.1008.1, 4.0.189.1, 4.0.432.1, 8.0.192036096.1, 8.0.192036096.2, 8.0.9144576.2

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 ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$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}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$