# Properties

 Label 16.16.321...808.1 Degree $16$ Signature $[16, 0]$ Discriminant $3.212\times 10^{29}$ Root discriminant $69.85$ Ramified primes $2, 3$ 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 - 48*x^14 + 936*x^12 - 9504*x^10 + 53604*x^8 - 166752*x^6 + 270864*x^4 - 202176*x^2 + 46818)

gp: K = bnfinit(x^16 - 48*x^14 + 936*x^12 - 9504*x^10 + 53604*x^8 - 166752*x^6 + 270864*x^4 - 202176*x^2 + 46818, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![46818, 0, -202176, 0, 270864, 0, -166752, 0, 53604, 0, -9504, 0, 936, 0, -48, 0, 1]);

$$x^{16} - 48 x^{14} + 936 x^{12} - 9504 x^{10} + 53604 x^{8} - 166752 x^{6} + 270864 x^{4} - 202176 x^{2} + 46818$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $16$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[16, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$321236373250909071617512439808$$$$\medspace = 2^{79}\cdot 3^{12}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $69.85$ 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: $2, 3$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{63} a^{8} - \frac{1}{21} a^{6} - \frac{1}{7} a^{4} + \frac{1}{7} a^{2} - \frac{2}{7}$, $\frac{1}{63} a^{9} - \frac{1}{21} a^{7} - \frac{1}{7} a^{5} + \frac{1}{7} a^{3} - \frac{2}{7} a$, $\frac{1}{63} a^{10} + \frac{1}{21} a^{6} + \frac{1}{21} a^{4} + \frac{1}{7} a^{2} + \frac{1}{7}$, $\frac{1}{63} a^{11} + \frac{1}{21} a^{7} + \frac{1}{21} a^{5} + \frac{1}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{189} a^{12} - \frac{1}{21} a^{6} - \frac{1}{7} a^{4} - \frac{3}{7} a^{2} + \frac{2}{7}$, $\frac{1}{3213} a^{13} + \frac{1}{357} a^{11} + \frac{1}{153} a^{9} - \frac{5}{357} a^{7} - \frac{1}{51} a^{5} + \frac{7}{17} a^{3} - \frac{44}{119} a$, $\frac{1}{3213} a^{14} - \frac{8}{3213} a^{12} + \frac{1}{153} a^{10} + \frac{2}{1071} a^{8} - \frac{1}{51} a^{6} + \frac{4}{51} a^{4} + \frac{24}{119} a^{2} + \frac{3}{7}$, $\frac{1}{3213} a^{15} - \frac{1}{357} a^{11} + \frac{1}{153} a^{9} - \frac{10}{119} a^{7} - \frac{4}{51} a^{5} - \frac{26}{119} a^{3} + \frac{5}{119} a$

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: $15$ 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: $$8601040814.68$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{16}\cdot(2\pi)^{0}\cdot 8601040814.68 \cdot 1}{2\sqrt{321236373250909071617512439808}}\approx 0.497265794585$ (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 R R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ $16$

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
2Data not computed
3Data not computed