# Properties

 Label 16.16.817...873.1 Degree $16$ Signature $[16, 0]$ Discriminant $8.175\times 10^{22}$ Root discriminant $27.04$ Ramified primes $13, 17$ 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 - 3*x^15 - 25*x^14 + 92*x^13 + 98*x^12 - 583*x^11 + 49*x^10 + 1383*x^9 - 647*x^8 - 1357*x^7 + 807*x^6 + 520*x^5 - 251*x^4 - 114*x^3 + 19*x^2 + 11*x + 1)

gp: K = bnfinit(x^16 - 3*x^15 - 25*x^14 + 92*x^13 + 98*x^12 - 583*x^11 + 49*x^10 + 1383*x^9 - 647*x^8 - 1357*x^7 + 807*x^6 + 520*x^5 - 251*x^4 - 114*x^3 + 19*x^2 + 11*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 11, 19, -114, -251, 520, 807, -1357, -647, 1383, 49, -583, 98, 92, -25, -3, 1]);

$$x^{16} - 3 x^{15} - 25 x^{14} + 92 x^{13} + 98 x^{12} - 583 x^{11} + 49 x^{10} + 1383 x^{9} - 647 x^{8} - 1357 x^{7} + 807 x^{6} + 520 x^{5} - 251 x^{4} - 114 x^{3} + 19 x^{2} + 11 x + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $16$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[16, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$81753664774171848863873$$$$\medspace = 13^{4}\cdot 17^{15}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $27.04$ 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: $13, 17$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{103} a^{14} - \frac{25}{103} a^{13} + \frac{9}{103} a^{12} + \frac{22}{103} a^{11} + \frac{17}{103} a^{10} + \frac{51}{103} a^{9} + \frac{43}{103} a^{8} - \frac{26}{103} a^{7} - \frac{15}{103} a^{6} + \frac{29}{103} a^{5} - \frac{22}{103} a^{4} + \frac{48}{103} a^{3} - \frac{49}{103} a^{2} - \frac{11}{103} a + \frac{1}{103}$, $\frac{1}{2549095397} a^{15} - \frac{7272992}{2549095397} a^{14} + \frac{1027924379}{2549095397} a^{13} + \frac{1194144945}{2549095397} a^{12} + \frac{78928993}{2549095397} a^{11} + \frac{1239857956}{2549095397} a^{10} + \frac{396383097}{2549095397} a^{9} + \frac{155290479}{2549095397} a^{8} + \frac{341311969}{2549095397} a^{7} + \frac{1017202377}{2549095397} a^{6} - \frac{769072344}{2549095397} a^{5} + \frac{1223619063}{2549095397} a^{4} + \frac{973861671}{2549095397} a^{3} - \frac{1270128219}{2549095397} a^{2} + \frac{1007307724}{2549095397} a + \frac{534884425}{2549095397}$

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: $$1330090.29679$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{16}\cdot(2\pi)^{0}\cdot 1330090.29679 \cdot 1}{2\sqrt{81753664774171848863873}}\approx 0.152432455359$ (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4} 13.4.2.2x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2} 13.4.0.1x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
17Data not computed