Properties

Label 16.0.4045039500390625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 11^{4}\cdot 29^{4}$
Root discriminant $9.45$
Ramified primes $5, 11, 29$
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, 0, 0, -3, 8, -11, 16, -27, 32, -14, -16, 27, -12, -6, 10, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 10*x^14 - 6*x^13 - 12*x^12 + 27*x^11 - 16*x^10 - 14*x^9 + 32*x^8 - 27*x^7 + 16*x^6 - 11*x^5 + 8*x^4 - 3*x^3 + 1)
gp: K = bnfinit(x^16 - 5*x^15 + 10*x^14 - 6*x^13 - 12*x^12 + 27*x^11 - 16*x^10 - 14*x^9 + 32*x^8 - 27*x^7 + 16*x^6 - 11*x^5 + 8*x^4 - 3*x^3 + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 5 x^{15} \) \(\mathstrut +\mathstrut 10 x^{14} \) \(\mathstrut -\mathstrut 6 x^{13} \) \(\mathstrut -\mathstrut 12 x^{12} \) \(\mathstrut +\mathstrut 27 x^{11} \) \(\mathstrut -\mathstrut 16 x^{10} \) \(\mathstrut -\mathstrut 14 x^{9} \) \(\mathstrut +\mathstrut 32 x^{8} \) \(\mathstrut -\mathstrut 27 x^{7} \) \(\mathstrut +\mathstrut 16 x^{6} \) \(\mathstrut -\mathstrut 11 x^{5} \) \(\mathstrut +\mathstrut 8 x^{4} \) \(\mathstrut -\mathstrut 3 x^{3} \) \(\mathstrut +\mathstrut 1 \)

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:  \(4045039500390625=5^{8}\cdot 11^{4}\cdot 29^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.45$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{67} a^{15} - \frac{3}{67} a^{14} + \frac{4}{67} a^{13} + \frac{2}{67} a^{12} - \frac{8}{67} a^{11} + \frac{11}{67} a^{10} + \frac{6}{67} a^{9} - \frac{2}{67} a^{8} + \frac{28}{67} a^{7} + \frac{29}{67} a^{6} + \frac{7}{67} a^{5} + \frac{3}{67} a^{4} + \frac{14}{67} a^{3} + \frac{25}{67} a^{2} - \frac{17}{67} a + \frac{33}{67}$

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{4}{67} a^{15} - \frac{12}{67} a^{14} + \frac{16}{67} a^{13} + \frac{8}{67} a^{12} - \frac{32}{67} a^{11} - \frac{23}{67} a^{10} + \frac{91}{67} a^{9} - \frac{8}{67} a^{8} - \frac{156}{67} a^{7} + \frac{116}{67} a^{6} + \frac{95}{67} a^{5} - \frac{189}{67} a^{4} + \frac{123}{67} a^{3} - \frac{101}{67} a^{2} + \frac{66}{67} a - \frac{2}{67} \),  \( \frac{70}{67} a^{15} - \frac{344}{67} a^{14} + \frac{548}{67} a^{13} + \frac{6}{67} a^{12} - \frac{1163}{67} a^{11} + \frac{1373}{67} a^{10} + \frac{85}{67} a^{9} - \frac{1547}{67} a^{8} + \frac{1491}{67} a^{7} - \frac{650}{67} a^{6} + \frac{289}{67} a^{5} - \frac{259}{67} a^{4} + \frac{176}{67} a^{3} - \frac{59}{67} a^{2} - \frac{51}{67} a - \frac{102}{67} \),  \( \frac{117}{67} a^{15} - \frac{418}{67} a^{14} + \frac{468}{67} a^{13} + \frac{301}{67} a^{12} - \frac{1271}{67} a^{11} + \frac{952}{67} a^{10} + \frac{568}{67} a^{9} - \frac{1440}{67} a^{8} + \frac{998}{67} a^{7} - \frac{426}{67} a^{6} + \frac{417}{67} a^{5} - \frac{453}{67} a^{4} + \frac{298}{67} a^{3} - \frac{23}{67} a^{2} + \frac{21}{67} a - \frac{92}{67} \),  \( \frac{92}{67} a^{15} - \frac{343}{67} a^{14} + \frac{368}{67} a^{13} + \frac{318}{67} a^{12} - \frac{1138}{67} a^{11} + \frac{744}{67} a^{10} + \frac{686}{67} a^{9} - \frac{1390}{67} a^{8} + \frac{767}{67} a^{7} - \frac{79}{67} a^{6} + \frac{108}{67} a^{5} - \frac{260}{67} a^{4} + \frac{149}{67} a^{3} + \frac{89}{67} a^{2} - \frac{23}{67} a - \frac{113}{67} \),  \( \frac{26}{67} a^{15} - \frac{11}{67} a^{14} - \frac{164}{67} a^{13} + \frac{320}{67} a^{12} - \frac{7}{67} a^{11} - \frac{585}{67} a^{10} + \frac{558}{67} a^{9} + \frac{149}{67} a^{8} - \frac{545}{67} a^{7} + \frac{352}{67} a^{6} - \frac{220}{67} a^{5} + \frac{279}{67} a^{4} - \frac{172}{67} a^{3} + \frac{47}{67} a^{2} + \frac{94}{67} a - \frac{13}{67} \),  \( \frac{71}{67} a^{15} - \frac{414}{67} a^{14} + \frac{820}{67} a^{13} - \frac{394}{67} a^{12} - \frac{1171}{67} a^{11} + \frac{2188}{67} a^{10} - \frac{914}{67} a^{9} - \frac{1482}{67} a^{8} + \frac{2457}{67} a^{7} - \frac{1760}{67} a^{6} + \frac{1033}{67} a^{5} - \frac{792}{67} a^{4} + \frac{525}{67} a^{3} - \frac{168}{67} a^{2} - \frac{135}{67} a - \frac{69}{67} \),  \( \frac{26}{67} a^{15} - \frac{212}{67} a^{14} + \frac{506}{67} a^{13} - \frac{350}{67} a^{12} - \frac{610}{67} a^{11} + \frac{1425}{67} a^{10} - \frac{782}{67} a^{9} - \frac{789}{67} a^{8} + \frac{1599}{67} a^{7} - \frac{1256}{67} a^{6} + \frac{718}{67} a^{5} - \frac{391}{67} a^{4} + \frac{163}{67} a^{3} - \frac{20}{67} a^{2} - \frac{107}{67} a - \frac{13}{67} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 8.56826776068 \)
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

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.2.275.1, 4.2.7975.1, 8.2.5781875.1 x2, 8.0.2193125.1 x2, 8.4.63600625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$