# Properties

 Label 16.0.92236816000...0000.1 Degree $16$ Signature $[0, 8]$ Discriminant $2^{16}\cdot 5^{12}\cdot 7^{8}$ Root discriminant $17.69$ Ramified primes $2, 5, 7$ Class number $1$ Class group Trivial 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![256, 0, 192, 0, 80, 0, 12, 0, -11, 0, 3, 0, 5, 0, 3, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3*x^14 + 5*x^12 + 3*x^10 - 11*x^8 + 12*x^6 + 80*x^4 + 192*x^2 + 256)

gp: K = bnfinit(x^16 + 3*x^14 + 5*x^12 + 3*x^10 - 11*x^8 + 12*x^6 + 80*x^4 + 192*x^2 + 256, 1)

## Normalizeddefining polynomial

$$x^{16} + 3 x^{14} + 5 x^{12} + 3 x^{10} - 11 x^{8} + 12 x^{6} + 80 x^{4} + 192 x^{2} + 256$$

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: $$92236816000000000000=2^{16}\cdot 5^{12}\cdot 7^{8}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $17.69$ magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 5, 7$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Gal(K/\Q)|$: $16$ This field is Galois and abelian over $\Q$. Conductor: $$140=2^{2}\cdot 5\cdot 7$$ Dirichlet character group: $\lbrace$$\chi_{140}(1,·), \chi_{140}(69,·), \chi_{140}(71,·), \chi_{140}(139,·), \chi_{140}(13,·), \chi_{140}(83,·), \chi_{140}(27,·), \chi_{140}(29,·), \chi_{140}(97,·), \chi_{140}(99,·), \chi_{140}(41,·), \chi_{140}(43,·), \chi_{140}(111,·), \chi_{140}(113,·), \chi_{140}(57,·), \chi_{140}(127,·)$$\rbrace$ This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{44} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{3}{11}$, $\frac{1}{88} a^{11} + \frac{1}{8} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{3}{22} a$, $\frac{1}{176} a^{12} - \frac{1}{176} a^{10} - \frac{5}{16} a^{8} - \frac{3}{16} a^{6} - \frac{5}{16} a^{4} + \frac{7}{22} a^{2} + \frac{2}{11}$, $\frac{1}{352} a^{13} - \frac{1}{352} a^{11} - \frac{5}{32} a^{9} - \frac{3}{32} a^{7} + \frac{11}{32} a^{5} - \frac{15}{44} a^{3} - \frac{9}{22} a$, $\frac{1}{704} a^{14} - \frac{1}{704} a^{12} - \frac{7}{704} a^{10} - \frac{19}{64} a^{8} + \frac{27}{64} a^{6} + \frac{7}{88} a^{4} + \frac{1}{22} a^{2} - \frac{2}{11}$, $\frac{1}{1408} a^{15} - \frac{1}{1408} a^{13} - \frac{7}{1408} a^{11} - \frac{19}{128} a^{9} + \frac{27}{128} a^{7} - \frac{81}{176} a^{5} - \frac{21}{44} a^{3} + \frac{9}{22} a$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

Trivial group, which has order $1$

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: $$-\frac{1}{88} a^{13} - \frac{1}{88} a^{3}$$ (order $20$) 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 magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$11381.1188025$$ 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4} 2.8.8.1x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2} 5.8.6.1x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4} 7.8.4.1x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$