# Properties

 Label 16.0.45086848686...0000.1 Degree $16$ Signature $[0, 8]$ Discriminant $2^{44}\cdot 3^{8}\cdot 5^{8}$ Root discriminant $26.05$ Ramified primes $2, 3, 5$ Class number $20$ (GRH) Class group $[20]$ (GRH) Galois group $C_4\times C_2^2$ (as 16T2)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, 96, 0, 456, 0, 624, 0, 608, 0, 312, 0, 114, 0, 12, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 12*x^14 + 114*x^12 + 312*x^10 + 608*x^8 + 624*x^6 + 456*x^4 + 96*x^2 + 16)

gp: K = bnfinit(x^16 + 12*x^14 + 114*x^12 + 312*x^10 + 608*x^8 + 624*x^6 + 456*x^4 + 96*x^2 + 16, 1)

## Normalizeddefining polynomial

$$x^{16} + 12 x^{14} + 114 x^{12} + 312 x^{10} + 608 x^{8} + 624 x^{6} + 456 x^{4} + 96 x^{2} + 16$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $16$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[0, 8]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$45086848686489600000000=2^{44}\cdot 3^{8}\cdot 5^{8}$$ magma: Discriminant(K);  sage: K.disc()  gp: K.disc Root discriminant: $26.05$ magma: Abs(Discriminant(K))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 3, 5$ magma: PrimeDivisors(Discriminant(K));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$240=2^{4}\cdot 3\cdot 5$$ Dirichlet character group: $\lbrace$$\chi_{240}(1,·), \chi_{240}(209,·), \chi_{240}(149,·), \chi_{240}(89,·), \chi_{240}(221,·), \chi_{240}(101,·), \chi_{240}(161,·), \chi_{240}(229,·), \chi_{240}(41,·), \chi_{240}(109,·), \chi_{240}(29,·), \chi_{240}(49,·), \chi_{240}(181,·), \chi_{240}(169,·), \chi_{240}(121,·), \chi_{240}(61,·)$$\rbrace$ This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{896} a^{12} - \frac{17}{112} a^{6} - \frac{55}{112}$, $\frac{1}{896} a^{13} - \frac{17}{112} a^{7} - \frac{55}{112} a$, $\frac{1}{27776} a^{14} - \frac{11}{27776} a^{12} - \frac{7}{62} a^{10} + \frac{375}{3472} a^{8} + \frac{131}{3472} a^{6} - \frac{6}{31} a^{4} + \frac{1625}{3472} a^{2} + \frac{829}{3472}$, $\frac{1}{27776} a^{15} - \frac{11}{27776} a^{13} - \frac{7}{62} a^{11} + \frac{375}{3472} a^{9} + \frac{131}{3472} a^{7} - \frac{6}{31} a^{5} + \frac{1625}{3472} a^{3} + \frac{829}{3472} a$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

$C_{20}$, which has order $20$ (assuming GRH)

magma: ClassGroup(K);

sage: K.class_group().invariants()

gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);

sage: UK = K.unit_group()

 Rank: $7$ magma: UnitRank(K);  sage: UK.rank()  gp: K.fu Torsion generator: $$-\frac{417}{27776} a^{14} - \frac{4899}{27776} a^{12} - \frac{207}{124} a^{10} - \frac{14891}{3472} a^{8} - \frac{28773}{3472} a^{6} - \frac{483}{62} a^{4} - \frac{21417}{3472} a^{2} - \frac{1035}{3472}$$ (order $6$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  sage: UK.torsion_generator()  gp: K.tu[2] 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) magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$12198.9512748$$ (assuming GRH) magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 An abelian group of order 16 The 16 conjugacy class representatives for $C_4\times C_2^2$ Character table for $C_4\times C_2^2$

## Intermediate fields

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

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];

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]])

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2} 2.8.22.2x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4} 3.8.4.1x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$