Properties

Label 16.0.8148125331379089.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 7^{6}\cdot 19^{4}$
Root discriminant $9.87$
Ramified primes $3, 7, 19$
Class number $1$
Class group Trivial
Galois Group $D_4^2.C_2$ (as 16T388)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -6, -4, 15, 6, -20, -7, 21, 7, -20, -6, 15, 4, -6, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 6*x^14 + 4*x^13 + 15*x^12 - 6*x^11 - 20*x^10 + 7*x^9 + 21*x^8 - 7*x^7 - 20*x^6 + 6*x^5 + 15*x^4 - 4*x^3 - 6*x^2 + x + 1)
gp: K = bnfinit(x^16 - x^15 - 6*x^14 + 4*x^13 + 15*x^12 - 6*x^11 - 20*x^10 + 7*x^9 + 21*x^8 - 7*x^7 - 20*x^6 + 6*x^5 + 15*x^4 - 4*x^3 - 6*x^2 + x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut x^{15} \) \(\mathstrut -\mathstrut 6 x^{14} \) \(\mathstrut +\mathstrut 4 x^{13} \) \(\mathstrut +\mathstrut 15 x^{12} \) \(\mathstrut -\mathstrut 6 x^{11} \) \(\mathstrut -\mathstrut 20 x^{10} \) \(\mathstrut +\mathstrut 7 x^{9} \) \(\mathstrut +\mathstrut 21 x^{8} \) \(\mathstrut -\mathstrut 7 x^{7} \) \(\mathstrut -\mathstrut 20 x^{6} \) \(\mathstrut +\mathstrut 6 x^{5} \) \(\mathstrut +\mathstrut 15 x^{4} \) \(\mathstrut -\mathstrut 4 x^{3} \) \(\mathstrut -\mathstrut 6 x^{2} \) \(\mathstrut +\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(8148125331379089=3^{12}\cdot 7^{6}\cdot 19^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.87$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} - \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{259} a^{15} - \frac{1}{37} a^{14} - \frac{38}{259} a^{13} + \frac{12}{37} a^{12} - \frac{45}{259} a^{11} + \frac{116}{259} a^{10} - \frac{13}{259} a^{9} - \frac{100}{259} a^{8} - \frac{82}{259} a^{7} + \frac{4}{259} a^{6} - \frac{118}{259} a^{5} + \frac{11}{259} a^{4} + \frac{23}{259} a^{3} - \frac{31}{259} a^{2} - \frac{5}{259} a - \frac{117}{259}$

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:  \( -\frac{1}{7} a^{15} + \frac{19}{7} a^{14} - \frac{8}{7} a^{13} - \frac{107}{7} a^{12} + \frac{27}{7} a^{11} + \frac{239}{7} a^{10} - \frac{13}{7} a^{9} - \frac{269}{7} a^{8} + \frac{27}{7} a^{7} + \frac{274}{7} a^{6} - \frac{27}{7} a^{5} - \frac{254}{7} a^{4} + \frac{15}{7} a^{3} + \frac{166}{7} a^{2} - \frac{13}{7} a - 5 \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{198}{259} a^{15} - \frac{239}{259} a^{14} - \frac{1086}{259} a^{13} + \frac{944}{259} a^{12} + \frac{2449}{259} a^{11} - \frac{181}{37} a^{10} - \frac{2944}{259} a^{9} + \frac{1068}{259} a^{8} + \frac{3004}{259} a^{7} - \frac{130}{37} a^{6} - \frac{2755}{259} a^{5} + \frac{772}{259} a^{4} + \frac{1927}{259} a^{3} - \frac{292}{259} a^{2} - \frac{509}{259} a - \frac{226}{259} \),  \( \frac{880}{259} a^{15} - \frac{1054}{259} a^{14} - \frac{4839}{259} a^{13} + \frac{4175}{259} a^{12} + \frac{11016}{259} a^{11} - \frac{6256}{259} a^{10} - \frac{12920}{259} a^{9} + \frac{7127}{259} a^{8} + \frac{12570}{259} a^{7} - \frac{7432}{259} a^{6} - \frac{12080}{259} a^{5} + \frac{6128}{259} a^{4} + \frac{7919}{259} a^{3} - \frac{4155}{259} a^{2} - \frac{1699}{259} a + \frac{102}{37} \),  \( \frac{393}{259} a^{15} - \frac{716}{259} a^{14} - \frac{257}{37} a^{13} + \frac{3190}{259} a^{12} + \frac{497}{37} a^{11} - \frac{5990}{259} a^{10} - \frac{3777}{259} a^{9} + \frac{1051}{37} a^{8} + \frac{3923}{259} a^{7} - \frac{7271}{259} a^{6} - \frac{3343}{259} a^{5} + \frac{6691}{259} a^{4} + \frac{1972}{259} a^{3} - \frac{4376}{259} a^{2} + \frac{33}{259} a + \frac{935}{259} \),  \( 6 a^{15} - \frac{43}{7} a^{14} - \frac{233}{7} a^{13} + \frac{160}{7} a^{12} + \frac{523}{7} a^{11} - \frac{225}{7} a^{10} - \frac{601}{7} a^{9} + \frac{281}{7} a^{8} + \frac{613}{7} a^{7} - \frac{267}{7} a^{6} - \frac{566}{7} a^{5} + \frac{225}{7} a^{4} + \frac{376}{7} a^{3} - \frac{153}{7} a^{2} - \frac{86}{7} a + \frac{29}{7} \),  \( \frac{71}{259} a^{15} - \frac{90}{259} a^{14} - \frac{330}{259} a^{13} + \frac{414}{259} a^{12} + \frac{468}{259} a^{11} - \frac{940}{259} a^{10} - \frac{5}{37} a^{9} + \frac{1558}{259} a^{8} + \frac{61}{259} a^{7} - \frac{1381}{259} a^{6} - \frac{34}{37} a^{5} + \frac{1151}{259} a^{4} - \frac{143}{259} a^{3} - \frac{1054}{259} a^{2} + \frac{459}{259} a + \frac{351}{259} \),  \( \frac{762}{259} a^{15} - \frac{709}{259} a^{14} - \frac{4425}{259} a^{13} + \frac{2588}{259} a^{12} + \frac{10443}{259} a^{11} - \frac{3331}{259} a^{10} - \frac{12718}{259} a^{9} + \frac{3868}{259} a^{8} + \frac{13033}{259} a^{7} - \frac{3723}{259} a^{6} - \frac{12179}{259} a^{5} + \frac{3239}{259} a^{4} + \frac{8387}{259} a^{3} - \frac{2088}{259} a^{2} - \frac{2330}{259} a + \frac{34}{37} \),  \( \frac{1075}{259} a^{15} - \frac{1013}{259} a^{14} - \frac{6070}{259} a^{13} + \frac{3572}{259} a^{12} + \frac{13859}{259} a^{11} - \frac{4504}{259} a^{10} - \frac{16084}{259} a^{9} + \frac{5646}{259} a^{8} + \frac{2334}{37} a^{7} - \frac{5505}{259} a^{6} - \frac{15258}{259} a^{5} + \frac{648}{37} a^{4} + \frac{10295}{259} a^{3} - \frac{400}{37} a^{2} - \frac{2452}{259} a + \frac{321}{259} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 41.5444168882 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_4^2.C_2$ (as 16T388):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 128
The 20 conjugacy class representatives for $D_4^2.C_2$
Character table for $D_4^2.C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.189.1, 4.0.1197.2, 4.0.513.1, 8.0.4750893.1 x2, 8.0.30088989.1 x2, 8.0.12895281.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{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$$[\ ]_{4}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$