# Properties

 Label 16.8.463...000.1 Degree $16$ Signature $[8, 4]$ Discriminant $4.635\times 10^{18}$ Root discriminant $14.68$ Ramified primes $2, 5, 29$ Class number $1$ (GRH) Class group trivial (GRH) 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 - 2*x^15 - 2*x^14 - 4*x^13 + 35*x^12 - 44*x^11 - 22*x^10 + 174*x^9 - 331*x^8 + 302*x^7 - 62*x^6 - 188*x^5 + 259*x^4 - 150*x^3 + 34*x^2 + 2*x - 1)

gp: K = bnfinit(x^16 - 2*x^15 - 2*x^14 - 4*x^13 + 35*x^12 - 44*x^11 - 22*x^10 + 174*x^9 - 331*x^8 + 302*x^7 - 62*x^6 - 188*x^5 + 259*x^4 - 150*x^3 + 34*x^2 + 2*x - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, 34, -150, 259, -188, -62, 302, -331, 174, -22, -44, 35, -4, -2, -2, 1]);

$$x^{16} - 2 x^{15} - 2 x^{14} - 4 x^{13} + 35 x^{12} - 44 x^{11} - 22 x^{10} + 174 x^{9} - 331 x^{8} + 302 x^{7} - 62 x^{6} - 188 x^{5} + 259 x^{4} - 150 x^{3} + 34 x^{2} + 2 x - 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $16$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[8, 4]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$4635236761600000000$$$$\medspace = 2^{24}\cdot 5^{8}\cdot 29^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $14.68$ 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: $2, 5, 29$ 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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{678271643933} a^{15} + \frac{325750529126}{678271643933} a^{14} - \frac{281507783993}{678271643933} a^{13} + \frac{6430993347}{23388677377} a^{12} - \frac{62961570293}{678271643933} a^{11} + \frac{157343054483}{678271643933} a^{10} - \frac{301370318331}{678271643933} a^{9} - \frac{10917822661}{52174741841} a^{8} + \frac{282971274632}{678271643933} a^{7} + \frac{228689566757}{678271643933} a^{6} - \frac{226933075903}{678271643933} a^{5} + \frac{105485188402}{678271643933} a^{4} - \frac{305553146047}{678271643933} a^{3} + \frac{4704375499}{678271643933} a^{2} - \frac{91615577874}{678271643933} a + \frac{191449920772}{678271643933}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $11$ 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 (assuming GRH) sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$1661.41008992$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 1661.41008992 \cdot 1}{2\sqrt{4635236761600000000}}\approx 0.153946679376$ (assuming GRH)

## 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 R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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
$2$Data not computed
$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}$
$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$$[\ ]^{2} 29.2.1.2x^{2} + 58$$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}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2} 29.2.1.2x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$