# 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)

# Related objects

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)

## Normalizeddefining 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

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ 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$$[\ ]_{2}^{4} 5.8.4.1x^{8} + 10 x^{6} + 125 x^{4} + 2500$$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$$[\ ]_{2}^{2} 11.4.0.1x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4} 11.4.2.1x^{4} + 143 x^{2} + 5929$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2} 29.2.1.2x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2} 29.2.1.2x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4} 29.4.0.1x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$