Properties

Label 16.0.5455162516701184.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 7^{4}\cdot 23^{2}$
Root discriminant $9.63$
Ramified primes $2, 7, 23$
Class number $1$
Class group Trivial
Galois Group $C_2\times D_4^2.C_2$ (as 16T509)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -4, 3, 0, 8, -8, 1, -8, 8, 0, 3, -4, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^13 + 3*x^12 + 8*x^10 - 8*x^9 + x^8 - 8*x^7 + 8*x^6 + 3*x^4 - 4*x^3 + 1)
gp: K = bnfinit(x^16 - 4*x^13 + 3*x^12 + 8*x^10 - 8*x^9 + x^8 - 8*x^7 + 8*x^6 + 3*x^4 - 4*x^3 + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 4 x^{13} \) \(\mathstrut +\mathstrut 3 x^{12} \) \(\mathstrut +\mathstrut 8 x^{10} \) \(\mathstrut -\mathstrut 8 x^{9} \) \(\mathstrut +\mathstrut x^{8} \) \(\mathstrut -\mathstrut 8 x^{7} \) \(\mathstrut +\mathstrut 8 x^{6} \) \(\mathstrut +\mathstrut 3 x^{4} \) \(\mathstrut -\mathstrut 4 x^{3} \) \(\mathstrut +\mathstrut 1 \)

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:  \(5455162516701184=2^{32}\cdot 7^{4}\cdot 23^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.63$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{13} + \frac{3}{7} a^{12} - \frac{1}{7} a^{11} + \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{49} a^{15} - \frac{1}{49} a^{14} - \frac{13}{49} a^{13} - \frac{12}{49} a^{12} + \frac{22}{49} a^{11} - \frac{8}{49} a^{10} - \frac{12}{49} a^{9} - \frac{3}{49} a^{8} - \frac{17}{49} a^{7} + \frac{23}{49} a^{6} + \frac{13}{49} a^{5} - \frac{20}{49} a^{4} - \frac{5}{49} a^{3} + \frac{15}{49} a^{2} - \frac{8}{49} a - \frac{13}{49}$

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:  \( -\frac{31}{49} a^{15} + \frac{59}{49} a^{14} + \frac{4}{49} a^{13} + \frac{113}{49} a^{12} - \frac{318}{49} a^{11} + \frac{157}{49} a^{10} - \frac{202}{49} a^{9} + \frac{625}{49} a^{8} - \frac{432}{49} a^{7} + \frac{211}{49} a^{6} - \frac{536}{49} a^{5} + \frac{284}{49} a^{4} - \frac{20}{49} a^{3} + \frac{158}{49} a^{2} - \frac{102}{49} a - \frac{10}{49} \) (order $8$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{2}{49} a^{15} + \frac{40}{49} a^{14} - \frac{12}{49} a^{13} + \frac{4}{49} a^{12} - \frac{145}{49} a^{11} + \frac{166}{49} a^{10} - \frac{52}{49} a^{9} + \frac{302}{49} a^{8} - \frac{370}{49} a^{7} + \frac{158}{49} a^{6} - \frac{296}{49} a^{5} + \frac{338}{49} a^{4} - \frac{52}{49} a^{3} + \frac{58}{49} a^{2} - \frac{100}{49} a + \frac{16}{49} \),  \( \frac{2}{49} a^{15} + \frac{40}{49} a^{14} - \frac{12}{49} a^{13} + \frac{4}{49} a^{12} - \frac{145}{49} a^{11} + \frac{166}{49} a^{10} - \frac{52}{49} a^{9} + \frac{302}{49} a^{8} - \frac{370}{49} a^{7} + \frac{158}{49} a^{6} - \frac{296}{49} a^{5} + \frac{289}{49} a^{4} - \frac{52}{49} a^{3} + \frac{58}{49} a^{2} - \frac{51}{49} a - \frac{33}{49} \),  \( \frac{13}{49} a^{15} + \frac{29}{49} a^{14} - \frac{8}{49} a^{13} - \frac{30}{49} a^{12} - \frac{50}{49} a^{11} + \frac{127}{49} a^{10} + \frac{12}{49} a^{9} + \frac{122}{49} a^{8} - \frac{214}{49} a^{7} + \frac{117}{49} a^{6} - \frac{153}{49} a^{5} + \frac{216}{49} a^{4} - \frac{58}{49} a^{3} + \frac{27}{49} a^{2} - \frac{90}{49} a + \frac{20}{49} \),  \( \frac{94}{49} a^{15} + \frac{4}{49} a^{14} + \frac{3}{49} a^{13} - \frac{344}{49} a^{12} + \frac{255}{49} a^{11} - \frac{17}{49} a^{10} + \frac{636}{49} a^{9} - \frac{576}{49} a^{8} + \frac{117}{49} a^{7} - \frac{582}{49} a^{6} + \frac{389}{49} a^{5} + \frac{31}{49} a^{4} + \frac{167}{49} a^{3} - \frac{109}{49} a^{2} - \frac{17}{49} a + \frac{3}{49} \),  \( \frac{67}{49} a^{15} + \frac{17}{49} a^{14} - \frac{10}{49} a^{13} - \frac{258}{49} a^{12} + \frac{116}{49} a^{11} + \frac{73}{49} a^{10} + \frac{463}{49} a^{9} - \frac{320}{49} a^{8} - \frac{145}{49} a^{7} - \frac{391}{49} a^{6} + \frac{227}{49} a^{5} + \frac{200}{49} a^{4} + \frac{71}{49} a^{3} - \frac{66}{49} a^{2} - \frac{67}{49} a + \frac{46}{49} \),  \( \frac{3}{7} a^{15} - \frac{4}{7} a^{14} - \frac{2}{7} a^{13} - \frac{11}{7} a^{12} + \frac{25}{7} a^{11} - \frac{5}{7} a^{10} + \frac{16}{7} a^{9} - 7 a^{8} + \frac{20}{7} a^{7} - \frac{13}{7} a^{6} + 6 a^{5} - \frac{6}{7} a^{4} - a^{3} - 2 a^{2} - \frac{1}{7} a + \frac{9}{7} \),  \( \frac{37}{49} a^{15} + \frac{47}{49} a^{14} - \frac{12}{49} a^{13} - \frac{143}{49} a^{12} - \frac{54}{49} a^{11} + \frac{166}{49} a^{10} + \frac{235}{49} a^{9} + \frac{15}{49} a^{8} - \frac{272}{49} a^{7} - \frac{150}{49} a^{6} - \frac{16}{49} a^{5} + \frac{163}{49} a^{4} + \frac{123}{49} a^{3} - \frac{26}{49} a^{2} - \frac{23}{49} a - \frac{5}{49} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 40.8921494432 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times D_4^2.C_2$ (as 16T509):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 256
The 40 conjugacy class representatives for $C_2\times D_4^2.C_2$
Character table for $C_2\times D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.2.1792.1, 4.2.448.1, \(\Q(\zeta_{8})\), 8.2.4616192.1, 8.2.73859072.3, 8.0.3211264.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$