# Properties

 Label 16.0.18014398509481984.1 Degree $16$ Signature $[0, 8]$ Discriminant $2^{54}$ Root discriminant $10.37$ Ramified prime $2$ Class number $1$ Class group Trivial Galois group $C_2^2 : C_4$ (as 16T10)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 104*x^13 + 220*x^12 - 368*x^11 + 516*x^10 - 624*x^9 + 664*x^8 - 624*x^7 + 516*x^6 - 368*x^5 + 220*x^4 - 104*x^3 + 36*x^2 - 8*x + 1)

gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 104*x^13 + 220*x^12 - 368*x^11 + 516*x^10 - 624*x^9 + 664*x^8 - 624*x^7 + 516*x^6 - 368*x^5 + 220*x^4 - 104*x^3 + 36*x^2 - 8*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 36, -104, 220, -368, 516, -624, 664, -624, 516, -368, 220, -104, 36, -8, 1]);

## Normalizeddefining polynomial

$$x^{16} - 8 x^{15} + 36 x^{14} - 104 x^{13} + 220 x^{12} - 368 x^{11} + 516 x^{10} - 624 x^{9} + 664 x^{8} - 624 x^{7} + 516 x^{6} - 368 x^{5} + 220 x^{4} - 104 x^{3} + 36 x^{2} - 8 x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $16$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 8]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$18014398509481984=2^{54}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $10.37$ 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: $2$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $16$ This field is Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} + \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{3}{7} a^{11} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{329} a^{15} - \frac{9}{329} a^{14} - \frac{2}{329} a^{13} - \frac{8}{329} a^{12} + \frac{87}{329} a^{11} + \frac{156}{329} a^{10} - \frac{110}{329} a^{9} - \frac{13}{47} a^{8} - \frac{138}{329} a^{7} - \frac{16}{329} a^{6} - \frac{79}{329} a^{5} + \frac{134}{329} a^{4} + \frac{39}{329} a^{3} - \frac{7}{47} a^{2} - \frac{9}{329} a + \frac{48}{329}$

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: $7$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-\frac{559}{329} a^{15} + \frac{4373}{329} a^{14} - \frac{2714}{47} a^{13} + \frac{52459}{329} a^{12} - \frac{104516}{329} a^{11} + \frac{164904}{329} a^{10} - \frac{218348}{329} a^{9} + \frac{250102}{329} a^{8} - \frac{251200}{329} a^{7} + \frac{222277}{329} a^{6} - \frac{169877}{329} a^{5} + \frac{15693}{47} a^{4} - \frac{55829}{329} a^{3} + \frac{2879}{47} a^{2} - \frac{3617}{329} a + \frac{240}{329}$$ (order $16$) 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: $$273.623706795$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$C_2^2:C_4$ (as 16T10):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 16 The 10 conjugacy class representatives for $C_2^2 : C_4$ Character table for $C_2^2 : C_4$

## Intermediate fields

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

## Sibling fields

 Degree 8 siblings: 8.4.67108864.1, 8.0.67108864.1

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed