# Properties

 Label 16.0.145...625.1 Degree $16$ Signature $[0, 8]$ Discriminant $1.458\times 10^{22}$ Root discriminant $24.28$ Ramified primes $5, 29, 89$ Class number $2$ Class group $[2]$ Galois group $D_4^2.C_2$ (as 16T388)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 25*x^14 - 73*x^13 + 165*x^12 - 268*x^11 + 424*x^10 - 728*x^9 + 889*x^8 - 430*x^7 + 441*x^6 - 2659*x^5 + 3672*x^4 - 2629*x^3 + 2709*x^2 - 260*x + 599)

gp: K = bnfinit(x^16 - 7*x^15 + 25*x^14 - 73*x^13 + 165*x^12 - 268*x^11 + 424*x^10 - 728*x^9 + 889*x^8 - 430*x^7 + 441*x^6 - 2659*x^5 + 3672*x^4 - 2629*x^3 + 2709*x^2 - 260*x + 599, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![599, -260, 2709, -2629, 3672, -2659, 441, -430, 889, -728, 424, -268, 165, -73, 25, -7, 1]);

$$x^{16} - 7 x^{15} + 25 x^{14} - 73 x^{13} + 165 x^{12} - 268 x^{11} + 424 x^{10} - 728 x^{9} + 889 x^{8} - 430 x^{7} + 441 x^{6} - 2659 x^{5} + 3672 x^{4} - 2629 x^{3} + 2709 x^{2} - 260 x + 599$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $16$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 8]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$14578339124454047265625$$$$\medspace = 5^{8}\cdot 29^{6}\cdot 89^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $24.28$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $5, 29, 89$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $8$ This field is not Galois over $\Q$. This is a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{15} a^{12} - \frac{1}{15} a^{11} + \frac{2}{15} a^{9} - \frac{1}{5} a^{8} + \frac{7}{15} a^{7} + \frac{2}{15} a^{6} + \frac{1}{15} a^{5} + \frac{4}{15} a^{4} - \frac{1}{3} a^{3} + \frac{2}{15} a^{2} - \frac{4}{15} a - \frac{4}{15}$, $\frac{1}{285} a^{13} - \frac{8}{285} a^{12} + \frac{1}{15} a^{11} + \frac{11}{285} a^{10} - \frac{59}{285} a^{9} + \frac{112}{285} a^{8} + \frac{127}{285} a^{7} + \frac{122}{285} a^{6} - \frac{27}{95} a^{5} + \frac{29}{95} a^{4} + \frac{124}{285} a^{3} + \frac{43}{95} a^{2} - \frac{7}{95} a - \frac{134}{285}$, $\frac{1}{285} a^{14} - \frac{7}{285} a^{12} + \frac{11}{285} a^{11} - \frac{28}{285} a^{10} + \frac{1}{285} a^{9} + \frac{18}{95} a^{8} - \frac{9}{19} a^{7} + \frac{59}{285} a^{6} - \frac{2}{57} a^{5} - \frac{18}{95} a^{4} + \frac{7}{15} a^{3} + \frac{118}{285} a^{2} + \frac{2}{285} a - \frac{47}{95}$, $\frac{1}{9624525387096911775} a^{15} - \frac{2221565146398242}{1924905077419382355} a^{14} - \frac{696746151088019}{641635025806460785} a^{13} + \frac{44653995410240977}{9624525387096911775} a^{12} + \frac{464975551791244669}{9624525387096911775} a^{11} + \frac{52712347408364486}{1924905077419382355} a^{10} - \frac{1267795370819490262}{3208175129032303925} a^{9} - \frac{166847647005016837}{1924905077419382355} a^{8} - \frac{727890951137860982}{3208175129032303925} a^{7} - \frac{1500118494758680717}{9624525387096911775} a^{6} - \frac{51371086828915627}{331880185761962475} a^{5} - \frac{56042006944074017}{384981015483876471} a^{4} - \frac{2474143894272142823}{9624525387096911775} a^{3} + \frac{563011741123746272}{1924905077419382355} a^{2} + \frac{239776087650287071}{506553967741942725} a - \frac{3758736368374675337}{9624525387096911775}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}$, which has order $2$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $7$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$13998.0176198$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 13998.0176198 \cdot 2}{2\sqrt{14578339124454047265625}}\approx 0.281612195459$

## Galois group

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

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 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

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 5.4.2.1x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 5.4.2.1x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2} 29.2.0.1x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2} 29.2.0.1x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4} 29.4.3.3x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2} 89.2.0.1x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2} 89.2.0.1x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 89.4.2.1x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$