Properties

Label 16.8.732...008.1
Degree $16$
Signature $[8, 4]$
Discriminant $7.328\times 10^{20}$
Root discriminant $20.14$
Ramified primes $2, 17$
Class number $1$
Class group trivial
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - x^14 + 21*x^13 - 33*x^12 + 30*x^11 - x^10 - 30*x^9 - 33*x^8 - 21*x^7 + 220*x^6 - 47*x^5 - 135*x^4 - 38*x^3 - x^2 + 4*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - x^14 + 21*x^13 - 33*x^12 + 30*x^11 - x^10 - 30*x^9 - 33*x^8 - 21*x^7 + 220*x^6 - 47*x^5 - 135*x^4 - 38*x^3 - x^2 + 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -1, -38, -135, -47, 220, -21, -33, -30, -1, 30, -33, 21, -1, -4, 1]);
 

\( x^{16} - 4 x^{15} - x^{14} + 21 x^{13} - 33 x^{12} + 30 x^{11} - x^{10} - 30 x^{9} - 33 x^{8} - 21 x^{7} + 220 x^{6} - 47 x^{5} - 135 x^{4} - 38 x^{3} - x^{2} + 4 x + 1 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(732780301186512843008\)\(\medspace = 2^{8}\cdot 17^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.14$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{101} a^{13} - \frac{19}{101} a^{12} + \frac{20}{101} a^{11} - \frac{21}{101} a^{10} - \frac{42}{101} a^{9} + \frac{25}{101} a^{8} + \frac{35}{101} a^{7} - \frac{27}{101} a^{6} + \frac{48}{101} a^{5} + \frac{43}{101} a^{4} + \frac{27}{101} a^{3} - \frac{10}{101} a^{2} - \frac{11}{101} a + \frac{46}{101}$, $\frac{1}{101} a^{14} - \frac{38}{101} a^{12} - \frac{45}{101} a^{11} - \frac{37}{101} a^{10} + \frac{35}{101} a^{9} + \frac{5}{101} a^{8} + \frac{32}{101} a^{7} + \frac{40}{101} a^{6} + \frac{46}{101} a^{5} + \frac{36}{101} a^{4} - \frac{2}{101} a^{3} + \frac{1}{101} a^{2} + \frac{39}{101} a - \frac{35}{101}$, $\frac{1}{4874751769} a^{15} + \frac{46136}{48264869} a^{14} + \frac{287264}{4874751769} a^{13} - \frac{243649810}{4874751769} a^{12} + \frac{173049271}{4874751769} a^{11} + \frac{161655478}{4874751769} a^{10} - \frac{2381238526}{4874751769} a^{9} + \frac{233598726}{4874751769} a^{8} + \frac{1003789055}{4874751769} a^{7} - \frac{2301253998}{4874751769} a^{6} - \frac{61689798}{4874751769} a^{5} - \frac{1402450380}{4874751769} a^{4} - \frac{1682431991}{4874751769} a^{3} + \frac{2211017123}{4874751769} a^{2} - \frac{2325954418}{4874751769} a - \frac{267146776}{4874751769}$

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

Class group and class number

Trivial group, which has order $1$

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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 23552.8271774 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 23552.8271774 \cdot 1}{2\sqrt{732780301186512843008}}\approx 0.173574369138$

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 R $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed