# Properties

 Label 16.8.346...273.1 Degree $16$ Signature $[8, 4]$ Discriminant $3.469\times 10^{25}$ Root discriminant $39.47$ Ramified primes $17, 59$ Class number $1$ (GRH) Class group trivial (GRH) 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 - x^15 + x^14 - 18*x^13 - 84*x^12 + 118*x^11 + 324*x^10 + 985*x^9 - 526*x^8 - 6172*x^7 - 1699*x^6 + 10250*x^5 + 14400*x^4 - 15760*x^3 - 20314*x^2 + 18495*x + 8263)

gp: K = bnfinit(x^16 - x^15 + x^14 - 18*x^13 - 84*x^12 + 118*x^11 + 324*x^10 + 985*x^9 - 526*x^8 - 6172*x^7 - 1699*x^6 + 10250*x^5 + 14400*x^4 - 15760*x^3 - 20314*x^2 + 18495*x + 8263, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8263, 18495, -20314, -15760, 14400, 10250, -1699, -6172, -526, 985, 324, 118, -84, -18, 1, -1, 1]);

$$x^{16} - x^{15} + x^{14} - 18 x^{13} - 84 x^{12} + 118 x^{11} + 324 x^{10} + 985 x^{9} - 526 x^{8} - 6172 x^{7} - 1699 x^{6} + 10250 x^{5} + 14400 x^{4} - 15760 x^{3} - 20314 x^{2} + 18495 x + 8263$$

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: $$34685013449866033007282273$$$$\medspace = 17^{15}\cdot 59^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $39.47$ 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: $17, 59$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{10847} a^{14} - \frac{4617}{10847} a^{13} - \frac{1235}{10847} a^{12} - \frac{1017}{10847} a^{11} - \frac{4615}{10847} a^{10} - \frac{2343}{10847} a^{9} - \frac{3837}{10847} a^{8} - \frac{2273}{10847} a^{7} - \frac{1356}{10847} a^{6} + \frac{914}{10847} a^{5} - \frac{15}{10847} a^{4} - \frac{4864}{10847} a^{3} - \frac{2371}{10847} a^{2} + \frac{482}{10847} a + \frac{1629}{10847}$, $\frac{1}{23066687042377216878374326477} a^{15} - \frac{907710944878488712988529}{23066687042377216878374326477} a^{14} + \frac{9902867254714521246335289779}{23066687042377216878374326477} a^{13} + \frac{5922616941654691369506466432}{23066687042377216878374326477} a^{12} - \frac{9710969246857321050997772932}{23066687042377216878374326477} a^{11} + \frac{587354455994741435999188293}{23066687042377216878374326477} a^{10} - \frac{4939423261427355964220622912}{23066687042377216878374326477} a^{9} - \frac{10011299109976230411961139432}{23066687042377216878374326477} a^{8} + \frac{8395861194049699040593603752}{23066687042377216878374326477} a^{7} - \frac{6377314445417707664551068229}{23066687042377216878374326477} a^{6} - \frac{6870531177674674029222107084}{23066687042377216878374326477} a^{5} - \frac{8976823118028577038633002487}{23066687042377216878374326477} a^{4} - \frac{8495547933487166287463515760}{23066687042377216878374326477} a^{3} - \frac{6879310562091791135995319415}{23066687042377216878374326477} a^{2} - \frac{8057752745861026680904674289}{23066687042377216878374326477} a + \frac{10666646382477627374711399322}{23066687042377216878374326477}$

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: $$4436153.77262$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 4436153.77262 \cdot 1}{2\sqrt{34685013449866033007282273}}\approx 0.150267513927$ (assuming GRH)

## 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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
17Data not computed
$59$59.8.0.1$x^{8} - x + 14$$1$$8$$0$$C_8$$[\ ]^{8} 59.8.4.2x^{8} - 205379 x^{2} + 169643054$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$