Properties

Label 16.0.5605632412826517.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{10}\cdot 37^{7}$
Root discriminant $9.64$
Ramified primes $3, 37$
Class number $1$
Class group Trivial
Galois Group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 14, -27, 43, -63, 86, -107, 121, -121, 104, -75, 46, -24, 11, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 75*x^11 + 104*x^10 - 121*x^9 + 121*x^8 - 107*x^7 + 86*x^6 - 63*x^5 + 43*x^4 - 27*x^3 + 14*x^2 - 5*x + 1)
gp: K = bnfinit(x^16 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 75*x^11 + 104*x^10 - 121*x^9 + 121*x^8 - 107*x^7 + 86*x^6 - 63*x^5 + 43*x^4 - 27*x^3 + 14*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 4 x^{15} \) \(\mathstrut +\mathstrut 11 x^{14} \) \(\mathstrut -\mathstrut 24 x^{13} \) \(\mathstrut +\mathstrut 46 x^{12} \) \(\mathstrut -\mathstrut 75 x^{11} \) \(\mathstrut +\mathstrut 104 x^{10} \) \(\mathstrut -\mathstrut 121 x^{9} \) \(\mathstrut +\mathstrut 121 x^{8} \) \(\mathstrut -\mathstrut 107 x^{7} \) \(\mathstrut +\mathstrut 86 x^{6} \) \(\mathstrut -\mathstrut 63 x^{5} \) \(\mathstrut +\mathstrut 43 x^{4} \) \(\mathstrut -\mathstrut 27 x^{3} \) \(\mathstrut +\mathstrut 14 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(5605632412826517=3^{10}\cdot 37^{7}\)
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{2}{13} a^{13} + \frac{6}{13} a^{12} + \frac{4}{13} a^{11} - \frac{6}{13} a^{10} + \frac{3}{13} a^{9} + \frac{3}{13} a^{8} + \frac{2}{13} a^{7} + \frac{4}{13} a^{6} + \frac{5}{13} a^{4} + \frac{6}{13} a^{3} - \frac{6}{13} a^{2} + \frac{4}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{15} + \frac{2}{13} a^{13} + \frac{5}{13} a^{12} - \frac{1}{13} a^{11} + \frac{2}{13} a^{10} - \frac{3}{13} a^{9} - \frac{4}{13} a^{8} + \frac{5}{13} a^{6} + \frac{5}{13} a^{5} - \frac{4}{13} a^{4} - \frac{5}{13} a^{3} + \frac{3}{13} a^{2} + \frac{2}{13} a + \frac{6}{13}$

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:  \( -\frac{14}{13} a^{15} + \frac{56}{13} a^{14} - \frac{150}{13} a^{13} + \frac{331}{13} a^{12} - \frac{620}{13} a^{11} + 78 a^{10} - \frac{1376}{13} a^{9} + \frac{1589}{13} a^{8} - \frac{1539}{13} a^{7} + \frac{1337}{13} a^{6} - \frac{1032}{13} a^{5} + \frac{752}{13} a^{4} - \frac{491}{13} a^{3} + \frac{298}{13} a^{2} - \frac{142}{13} a + \frac{47}{13} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{4}{13} a^{14} - \frac{5}{13} a^{13} + \frac{24}{13} a^{12} - \frac{36}{13} a^{11} + \frac{80}{13} a^{10} - \frac{105}{13} a^{9} + \frac{155}{13} a^{8} - \frac{161}{13} a^{7} + \frac{172}{13} a^{6} - 10 a^{5} + \frac{111}{13} a^{4} - \frac{80}{13} a^{3} + \frac{54}{13} a^{2} - \frac{23}{13} a + \frac{14}{13} \),  \( \frac{48}{13} a^{15} - \frac{153}{13} a^{14} + \frac{388}{13} a^{13} - \frac{795}{13} a^{12} + \frac{1459}{13} a^{11} - 170 a^{10} + \frac{2829}{13} a^{9} - \frac{2978}{13} a^{8} + \frac{2736}{13} a^{7} - \frac{2257}{13} a^{6} + \frac{1696}{13} a^{5} - \frac{1152}{13} a^{4} + \frac{766}{13} a^{3} - \frac{446}{13} a^{2} + \frac{160}{13} a - \frac{33}{13} \),  \( \frac{8}{13} a^{15} - \frac{8}{13} a^{14} + a^{13} + \frac{5}{13} a^{12} - \frac{27}{13} a^{11} + \frac{129}{13} a^{10} - \frac{269}{13} a^{9} + \frac{438}{13} a^{8} - \frac{497}{13} a^{7} + \frac{463}{13} a^{6} - \frac{376}{13} a^{5} + \frac{292}{13} a^{4} - \frac{205}{13} a^{3} + \frac{137}{13} a^{2} - \frac{94}{13} a + \frac{33}{13} \),  \( \frac{9}{13} a^{15} - \frac{36}{13} a^{14} + \frac{89}{13} a^{13} - \frac{184}{13} a^{12} + \frac{341}{13} a^{11} - 40 a^{10} + \frac{671}{13} a^{9} - \frac{690}{13} a^{8} + \frac{617}{13} a^{7} - \frac{489}{13} a^{6} + \frac{383}{13} a^{5} - \frac{255}{13} a^{4} + \frac{168}{13} a^{3} - \frac{82}{13} a^{2} + \frac{17}{13} a + \frac{6}{13} \),  \( \frac{24}{13} a^{15} - \frac{62}{13} a^{14} + \frac{145}{13} a^{13} - \frac{278}{13} a^{12} + \frac{482}{13} a^{11} - \frac{659}{13} a^{10} + \frac{743}{13} a^{9} - \frac{646}{13} a^{8} + \frac{513}{13} a^{7} - \frac{375}{13} a^{6} + \frac{263}{13} a^{5} - \frac{146}{13} a^{4} + \frac{106}{13} a^{3} - \frac{24}{13} a^{2} - \frac{18}{13} a + \frac{18}{13} \),  \( \frac{24}{13} a^{15} - \frac{56}{13} a^{14} + \frac{144}{13} a^{13} - \frac{268}{13} a^{12} + \frac{480}{13} a^{11} - \frac{656}{13} a^{10} + \frac{774}{13} a^{9} - \frac{706}{13} a^{8} + \frac{590}{13} a^{7} - 33 a^{6} + \frac{276}{13} a^{5} - \frac{155}{13} a^{4} + \frac{90}{13} a^{3} - \frac{21}{13} a^{2} - \frac{20}{13} a + 1 \),  \( \frac{20}{13} a^{15} - \frac{70}{13} a^{14} + \frac{173}{13} a^{13} - \frac{359}{13} a^{12} + \frac{649}{13} a^{11} - \frac{996}{13} a^{10} + \frac{1251}{13} a^{9} - \frac{1304}{13} a^{8} + \frac{1147}{13} a^{7} - \frac{934}{13} a^{6} + \frac{685}{13} a^{5} - \frac{469}{13} a^{4} + 23 a^{3} - \frac{183}{13} a^{2} + \frac{59}{13} a - \frac{8}{13} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 32.1900879299 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.333.1, 8.0.4102893.1

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 $16$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $16$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$37$37.4.3.4$x^{4} + 296$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$