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Properties

Label 16.0.62126403993...8496.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{4}\cdot 17^{8}$
Root discriminant $30.70$
Ramified primes $2, 3, 17$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $D_8:C_2$ (as 16T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![178, 40, -432, 32, 794, -100, -448, 272, 347, -100, 360, -40, 142, -8, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 20*x^14 - 8*x^13 + 142*x^12 - 40*x^11 + 360*x^10 - 100*x^9 + 347*x^8 + 272*x^7 - 448*x^6 - 100*x^5 + 794*x^4 + 32*x^3 - 432*x^2 + 40*x + 178)
 
gp: K = bnfinit(x^16 + 20*x^14 - 8*x^13 + 142*x^12 - 40*x^11 + 360*x^10 - 100*x^9 + 347*x^8 + 272*x^7 - 448*x^6 - 100*x^5 + 794*x^4 + 32*x^3 - 432*x^2 + 40*x + 178, 1)
 

Normalized defining polynomial

\( x^{16} + 20 x^{14} - 8 x^{13} + 142 x^{12} - 40 x^{11} + 360 x^{10} - 100 x^{9} + 347 x^{8} + 272 x^{7} - 448 x^{6} - 100 x^{5} + 794 x^{4} + 32 x^{3} - 432 x^{2} + 40 x + 178 \)

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:  \(621264039932880796778496=2^{40}\cdot 3^{4}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.70$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{69} a^{14} + \frac{5}{69} a^{13} - \frac{5}{69} a^{12} + \frac{16}{69} a^{11} - \frac{34}{69} a^{10} + \frac{25}{69} a^{9} + \frac{7}{69} a^{7} - \frac{32}{69} a^{6} - \frac{31}{69} a^{5} - \frac{5}{23} a^{4} + \frac{6}{23} a^{3} + \frac{1}{69} a^{2} - \frac{4}{23} a + \frac{10}{69}$, $\frac{1}{10307712791163634204857} a^{15} + \frac{20485970235494054576}{10307712791163634204857} a^{14} + \frac{274292578608582898794}{3435904263721211401619} a^{13} - \frac{261324612864912502950}{3435904263721211401619} a^{12} + \frac{4540521834661737695510}{10307712791163634204857} a^{11} + \frac{1462934999794107752770}{3435904263721211401619} a^{10} - \frac{3824938275402015489280}{10307712791163634204857} a^{9} + \frac{1309688875274001500122}{10307712791163634204857} a^{8} - \frac{1432074531051919207529}{10307712791163634204857} a^{7} - \frac{4292830243195200646412}{10307712791163634204857} a^{6} + \frac{988700399896404512898}{3435904263721211401619} a^{5} - \frac{1181915591764617507074}{10307712791163634204857} a^{4} - \frac{771124349860744189692}{3435904263721211401619} a^{3} - \frac{2882530916725652533460}{10307712791163634204857} a^{2} + \frac{724321472707007144009}{10307712791163634204857} a - \frac{88198518250078310065}{10307712791163634204857}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (assuming GRH)

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:  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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 45197.9459907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8:C_2$ (as 16T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.4.4352.1 x2, 4.4.9248.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.5473632256.1

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

Sibling fields

Galois closure: data not computed
Degree 16 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.20.53$x^{8} + 4 x^{5} + 2 x^{4} + 2$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
2.8.20.53$x^{8} + 4 x^{5} + 2 x^{4} + 2$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$