# Properties

 Label 16.0.3100665090835456.1 Degree $16$ Signature $[0, 8]$ Discriminant $2^{10}\cdot 1740113^{2}$ Root discriminant $9.29$ Ramified primes $2, 1740113$ Class number $1$ Class group Trivial Galois Group 16T1945

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 3, 0, 5, 0, 7, 0, 8, 0, 6, 0, 4, 0, 2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 + 4*x^12 + 6*x^10 + 8*x^8 + 7*x^6 + 5*x^4 + 3*x^2 + 1)
gp: K = bnfinit(x^16 + 2*x^14 + 4*x^12 + 6*x^10 + 8*x^8 + 7*x^6 + 5*x^4 + 3*x^2 + 1, 1)

## Normalizeddefining polynomial

$$x^{16}$$ $$\mathstrut +\mathstrut 2 x^{14}$$ $$\mathstrut +\mathstrut 4 x^{12}$$ $$\mathstrut +\mathstrut 6 x^{10}$$ $$\mathstrut +\mathstrut 8 x^{8}$$ $$\mathstrut +\mathstrut 7 x^{6}$$ $$\mathstrut +\mathstrut 5 x^{4}$$ $$\mathstrut +\mathstrut 3 x^{2}$$ $$\mathstrut +\mathstrut 1$$

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: $$3100665090835456=2^{10}\cdot 1740113^{2}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $9.29$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 1740113$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

## Class group and class number

Trivial Abelian group, 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: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$a$$,  $$a^{15} + a^{13} + 2 a^{11} + 2 a^{9} + 3 a^{7}$$,  $$a^{12} + a^{10} + 3 a^{8} + 3 a^{6} + 4 a^{4} + 2 a^{2} + 1$$,  $$\frac{1}{2} a^{14} + \frac{1}{2} a^{13} + \frac{1}{2} a^{12} + a^{11} + \frac{3}{2} a^{10} + 2 a^{9} + 2 a^{8} + \frac{5}{2} a^{7} + \frac{5}{2} a^{6} + \frac{7}{2} a^{5} + \frac{3}{2} a^{4} + 2 a^{3} + 2 a^{2} + \frac{3}{2} a + 1$$,  $$\frac{1}{2} a^{15} + \frac{1}{2} a^{14} + \frac{1}{2} a^{13} + a^{12} + \frac{1}{2} a^{11} + 2 a^{10} + a^{9} + \frac{5}{2} a^{8} + \frac{1}{2} a^{7} + \frac{7}{2} a^{6} - \frac{1}{2} a^{5} + 2 a^{4} - a^{3} + \frac{3}{2} a^{2}$$,  $$\frac{3}{2} a^{15} + a^{14} + \frac{3}{2} a^{13} + 2 a^{12} + 4 a^{11} + \frac{7}{2} a^{10} + \frac{9}{2} a^{9} + \frac{11}{2} a^{8} + 6 a^{7} + \frac{13}{2} a^{6} + \frac{5}{2} a^{5} + 5 a^{4} + \frac{5}{2} a^{3} + \frac{5}{2} a^{2} + \frac{1}{2} a + \frac{3}{2}$$,  $$\frac{1}{2} a^{15} - a^{14} + \frac{1}{2} a^{13} - a^{12} + 2 a^{11} - \frac{5}{2} a^{10} + \frac{5}{2} a^{9} - \frac{5}{2} a^{8} + 4 a^{7} - \frac{7}{2} a^{6} + \frac{7}{2} a^{5} - a^{4} + \frac{7}{2} a^{3} - \frac{1}{2} a^{2} + \frac{3}{2} a - \frac{1}{2}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$7.31402076949$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A non-solvable group of order 5160960 The 100 conjugacy class representatives for t16n1945 are not computed Character table for t16n1945 is not computed

## Intermediate fields

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

## Sibling fields

 Degree 16 sibling: data not computed Degree 32 sibling: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6} 2.10.10.4x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
1740113Data not computed