Properties

Label 16.0.5572562780160000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{4}$
Root discriminant $9.64$
Ramified primes $2, 3, 5$
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, 0, 8, 0, 32, 0, 50, 0, 34, 0, 4, 0, -7, 0, -2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 - 7*x^12 + 4*x^10 + 34*x^8 + 50*x^6 + 32*x^4 + 8*x^2 + 1)
gp: K = bnfinit(x^16 - 2*x^14 - 7*x^12 + 4*x^10 + 34*x^8 + 50*x^6 + 32*x^4 + 8*x^2 + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut -\mathstrut 7 x^{12} \) \(\mathstrut +\mathstrut 4 x^{10} \) \(\mathstrut +\mathstrut 34 x^{8} \) \(\mathstrut +\mathstrut 50 x^{6} \) \(\mathstrut +\mathstrut 32 x^{4} \) \(\mathstrut +\mathstrut 8 x^{2} \) \(\mathstrut +\mathstrut 1 \)

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:  \(5572562780160000=2^{24}\cdot 3^{12}\cdot 5^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.64$
magma: Abs(Discriminant(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(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{572} a^{14} + \frac{43}{572} a^{12} - \frac{1}{4} a^{11} + \frac{69}{572} a^{10} + \frac{53}{286} a^{8} - \frac{1}{4} a^{7} + \frac{57}{143} a^{6} + \frac{7}{286} a^{4} - \frac{1}{4} a^{3} + \frac{233}{572} a^{2} - \frac{1}{4} a + \frac{197}{572}$, $\frac{1}{572} a^{15} + \frac{43}{572} a^{13} - \frac{37}{286} a^{11} - \frac{1}{4} a^{10} - \frac{37}{572} a^{9} - \frac{29}{286} a^{7} - \frac{1}{4} a^{6} - \frac{68}{143} a^{5} - \frac{49}{143} a^{3} - \frac{1}{4} a^{2} + \frac{197}{572} a - \frac{1}{4}$

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

Class group and class number

Trivial Abelian group, 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:  \( \frac{19}{143} a^{14} - \frac{41}{143} a^{12} - \frac{119}{143} a^{10} + \frac{167}{286} a^{8} + \frac{1085}{286} a^{6} + \frac{981}{143} a^{4} + \frac{709}{143} a^{2} + \frac{479}{286} \) (order $12$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{136}{143} a^{15} + \frac{67}{286} a^{14} - \frac{1347}{572} a^{13} - \frac{61}{143} a^{12} - \frac{769}{143} a^{11} - \frac{525}{286} a^{10} + \frac{831}{143} a^{9} + \frac{619}{572} a^{8} + \frac{16639}{572} a^{7} + \frac{4669}{572} a^{6} + \frac{20343}{572} a^{5} + \frac{1756}{143} a^{4} + \frac{9635}{572} a^{3} + \frac{2169}{286} a^{2} + \frac{633}{572} a + \frac{801}{572} \),  \( \frac{56}{143} a^{15} - \frac{5}{44} a^{14} - \frac{166}{143} a^{13} + \frac{5}{44} a^{12} - \frac{989}{572} a^{11} + \frac{51}{44} a^{10} + \frac{502}{143} a^{9} - \frac{1}{22} a^{8} + \frac{6027}{572} a^{7} - \frac{54}{11} a^{6} + \frac{1213}{143} a^{5} - \frac{189}{22} a^{4} + \frac{855}{572} a^{3} - \frac{241}{44} a^{2} - \frac{917}{572} a - \frac{17}{44} \),  \( \frac{189}{286} a^{15} - \frac{71}{286} a^{14} - \frac{763}{572} a^{13} + \frac{329}{572} a^{12} - \frac{2661}{572} a^{11} + \frac{927}{572} a^{10} + \frac{1601}{572} a^{9} - \frac{519}{286} a^{8} + \frac{13111}{572} a^{7} - \frac{1087}{143} a^{6} + \frac{18591}{572} a^{5} - \frac{5563}{572} a^{4} + \frac{5141}{286} a^{3} - \frac{621}{143} a^{2} + \frac{1393}{572} a - \frac{58}{143} \),  \( \frac{49}{286} a^{15} + \frac{5}{44} a^{14} - \frac{181}{286} a^{13} - \frac{5}{44} a^{12} - \frac{245}{572} a^{11} - \frac{51}{44} a^{10} + \frac{309}{143} a^{9} + \frac{1}{22} a^{8} + \frac{2181}{572} a^{7} + \frac{54}{11} a^{6} + \frac{57}{143} a^{5} + \frac{189}{22} a^{4} - \frac{2477}{572} a^{3} + \frac{241}{44} a^{2} - \frac{1429}{572} a + \frac{61}{44} \),  \( \frac{423}{572} a^{15} + \frac{17}{286} a^{14} - \frac{973}{572} a^{13} + \frac{8}{143} a^{12} - \frac{1351}{286} a^{11} - \frac{657}{572} a^{10} + \frac{2653}{572} a^{9} + \frac{43}{143} a^{8} + \frac{3376}{143} a^{7} + \frac{2175}{572} a^{6} + \frac{8395}{286} a^{5} + \frac{1525}{286} a^{4} + \frac{4163}{286} a^{3} + \frac{1201}{572} a^{2} + \frac{677}{572} a - \frac{23}{572} \),  \( \frac{61}{143} a^{15} + \frac{123}{572} a^{14} - \frac{519}{572} a^{13} - \frac{145}{572} a^{12} - \frac{1611}{572} a^{11} - \frac{1237}{572} a^{10} + \frac{491}{286} a^{9} + \frac{311}{572} a^{8} + \frac{4221}{286} a^{7} + \frac{5307}{572} a^{6} + \frac{11567}{572} a^{5} + \frac{4007}{286} a^{4} + \frac{1486}{143} a^{3} + \frac{4635}{572} a^{2} + \frac{5}{143} a + \frac{159}{143} \),  \( \frac{128}{143} a^{15} - \frac{575}{286} a^{13} - \frac{1641}{286} a^{11} + \frac{698}{143} a^{9} + \frac{4159}{143} a^{7} + \frac{10877}{286} a^{5} + \frac{2940}{143} a^{3} + \frac{334}{143} a \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 62.6647388247 \)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.0.320.1, 4.0.2880.1, 4.0.432.1 x2, 4.2.1728.1 x2, \(\Q(\zeta_{12})\), 8.0.2985984.1, 8.0.4665600.1, 8.0.8294400.2

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 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
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}$