Properties

Label 16.8.346...273.1
Degree $16$
Signature $[8, 4]$
Discriminant $3.469\times 10^{25}$
Root discriminant $39.47$
Ramified primes $17, 59$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^14 - 18*x^13 - 84*x^12 + 118*x^11 + 324*x^10 + 985*x^9 - 526*x^8 - 6172*x^7 - 1699*x^6 + 10250*x^5 + 14400*x^4 - 15760*x^3 - 20314*x^2 + 18495*x + 8263)
 
gp: K = bnfinit(x^16 - x^15 + x^14 - 18*x^13 - 84*x^12 + 118*x^11 + 324*x^10 + 985*x^9 - 526*x^8 - 6172*x^7 - 1699*x^6 + 10250*x^5 + 14400*x^4 - 15760*x^3 - 20314*x^2 + 18495*x + 8263, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8263, 18495, -20314, -15760, 14400, 10250, -1699, -6172, -526, 985, 324, 118, -84, -18, 1, -1, 1]);
 

\( x^{16} - x^{15} + x^{14} - 18 x^{13} - 84 x^{12} + 118 x^{11} + 324 x^{10} + 985 x^{9} - 526 x^{8} - 6172 x^{7} - 1699 x^{6} + 10250 x^{5} + 14400 x^{4} - 15760 x^{3} - 20314 x^{2} + 18495 x + 8263 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(34685013449866033007282273\)\(\medspace = 17^{15}\cdot 59^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $39.47$
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:  $17, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{10847} a^{14} - \frac{4617}{10847} a^{13} - \frac{1235}{10847} a^{12} - \frac{1017}{10847} a^{11} - \frac{4615}{10847} a^{10} - \frac{2343}{10847} a^{9} - \frac{3837}{10847} a^{8} - \frac{2273}{10847} a^{7} - \frac{1356}{10847} a^{6} + \frac{914}{10847} a^{5} - \frac{15}{10847} a^{4} - \frac{4864}{10847} a^{3} - \frac{2371}{10847} a^{2} + \frac{482}{10847} a + \frac{1629}{10847}$, $\frac{1}{23066687042377216878374326477} a^{15} - \frac{907710944878488712988529}{23066687042377216878374326477} a^{14} + \frac{9902867254714521246335289779}{23066687042377216878374326477} a^{13} + \frac{5922616941654691369506466432}{23066687042377216878374326477} a^{12} - \frac{9710969246857321050997772932}{23066687042377216878374326477} a^{11} + \frac{587354455994741435999188293}{23066687042377216878374326477} a^{10} - \frac{4939423261427355964220622912}{23066687042377216878374326477} a^{9} - \frac{10011299109976230411961139432}{23066687042377216878374326477} a^{8} + \frac{8395861194049699040593603752}{23066687042377216878374326477} a^{7} - \frac{6377314445417707664551068229}{23066687042377216878374326477} a^{6} - \frac{6870531177674674029222107084}{23066687042377216878374326477} a^{5} - \frac{8976823118028577038633002487}{23066687042377216878374326477} a^{4} - \frac{8495547933487166287463515760}{23066687042377216878374326477} a^{3} - \frac{6879310562091791135995319415}{23066687042377216878374326477} a^{2} - \frac{8057752745861026680904674289}{23066687042377216878374326477} a + \frac{10666646382477627374711399322}{23066687042377216878374326477}$

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:  $11$
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:  \( 4436153.77262 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 4436153.77262 \cdot 1}{2\sqrt{34685013449866033007282273}}\approx 0.150267513927$ (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

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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
17Data not computed
$59$59.8.0.1$x^{8} - x + 14$$1$$8$$0$$C_8$$[\ ]^{8}$
59.8.4.2$x^{8} - 205379 x^{2} + 169643054$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$