# Properties

 Label 16.8.732...008.1 Degree $16$ Signature $[8, 4]$ Discriminant $7.328\times 10^{20}$ Root discriminant $20.14$ Ramified primes $2, 17$ Class number $1$ Class group trivial Galois group $C_{16} : C_2$ (as 16T22)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - x^14 + 21*x^13 - 33*x^12 + 30*x^11 - x^10 - 30*x^9 - 33*x^8 - 21*x^7 + 220*x^6 - 47*x^5 - 135*x^4 - 38*x^3 - x^2 + 4*x + 1)

gp: K = bnfinit(x^16 - 4*x^15 - x^14 + 21*x^13 - 33*x^12 + 30*x^11 - x^10 - 30*x^9 - 33*x^8 - 21*x^7 + 220*x^6 - 47*x^5 - 135*x^4 - 38*x^3 - x^2 + 4*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -1, -38, -135, -47, 220, -21, -33, -30, -1, 30, -33, 21, -1, -4, 1]);

$$x^{16} - 4 x^{15} - x^{14} + 21 x^{13} - 33 x^{12} + 30 x^{11} - x^{10} - 30 x^{9} - 33 x^{8} - 21 x^{7} + 220 x^{6} - 47 x^{5} - 135 x^{4} - 38 x^{3} - x^{2} + 4 x + 1$$

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: $$732780301186512843008$$$$\medspace = 2^{8}\cdot 17^{15}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $20.14$ 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: $2, 17$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{101} a^{13} - \frac{19}{101} a^{12} + \frac{20}{101} a^{11} - \frac{21}{101} a^{10} - \frac{42}{101} a^{9} + \frac{25}{101} a^{8} + \frac{35}{101} a^{7} - \frac{27}{101} a^{6} + \frac{48}{101} a^{5} + \frac{43}{101} a^{4} + \frac{27}{101} a^{3} - \frac{10}{101} a^{2} - \frac{11}{101} a + \frac{46}{101}$, $\frac{1}{101} a^{14} - \frac{38}{101} a^{12} - \frac{45}{101} a^{11} - \frac{37}{101} a^{10} + \frac{35}{101} a^{9} + \frac{5}{101} a^{8} + \frac{32}{101} a^{7} + \frac{40}{101} a^{6} + \frac{46}{101} a^{5} + \frac{36}{101} a^{4} - \frac{2}{101} a^{3} + \frac{1}{101} a^{2} + \frac{39}{101} a - \frac{35}{101}$, $\frac{1}{4874751769} a^{15} + \frac{46136}{48264869} a^{14} + \frac{287264}{4874751769} a^{13} - \frac{243649810}{4874751769} a^{12} + \frac{173049271}{4874751769} a^{11} + \frac{161655478}{4874751769} a^{10} - \frac{2381238526}{4874751769} a^{9} + \frac{233598726}{4874751769} a^{8} + \frac{1003789055}{4874751769} a^{7} - \frac{2301253998}{4874751769} a^{6} - \frac{61689798}{4874751769} a^{5} - \frac{1402450380}{4874751769} a^{4} - \frac{1682431991}{4874751769} a^{3} + \frac{2211017123}{4874751769} a^{2} - \frac{2325954418}{4874751769} a - \frac{267146776}{4874751769}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

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 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$23552.8271774$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 23552.8271774 \cdot 1}{2\sqrt{732780301186512843008}}\approx 0.173574369138$

## Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 32 The 20 conjugacy class representatives for $C_{16} : C_2$ Character table for $C_{16} : C_2$

## Intermediate fields

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

## Sibling fields

 Galois closure: Deg 32

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4} 2.8.0.1x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed