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Properties

Label 16.0.34789235097...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{4}\cdot 5^{8}$
Root discriminant $16.65$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $Q_8 : C_2^2$ (as 16T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -84, 320, -612, 526, 60, -460, 188, 282, -364, 128, 60, -70, 20, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 20*x^13 - 70*x^12 + 60*x^11 + 128*x^10 - 364*x^9 + 282*x^8 + 188*x^7 - 460*x^6 + 60*x^5 + 526*x^4 - 612*x^3 + 320*x^2 - 84*x + 9)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 + 20*x^13 - 70*x^12 + 60*x^11 + 128*x^10 - 364*x^9 + 282*x^8 + 188*x^7 - 460*x^6 + 60*x^5 + 526*x^4 - 612*x^3 + 320*x^2 - 84*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} + 20 x^{13} - 70 x^{12} + 60 x^{11} + 128 x^{10} - 364 x^{9} + 282 x^{8} + 188 x^{7} - 460 x^{6} + 60 x^{5} + 526 x^{4} - 612 x^{3} + 320 x^{2} - 84 x + 9 \)

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:  \(34789235097600000000=2^{40}\cdot 3^{4}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.65$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{18} a^{11} + \frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{2}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{5}{18} a^{5} + \frac{7}{18} a^{4} + \frac{7}{18} a^{3} + \frac{5}{18} a^{2} - \frac{1}{6} a$, $\frac{1}{126} a^{14} + \frac{1}{42} a^{13} + \frac{1}{126} a^{12} - \frac{5}{63} a^{11} - \frac{1}{42} a^{10} - \frac{4}{63} a^{9} + \frac{11}{126} a^{8} - \frac{10}{21} a^{7} - \frac{17}{126} a^{6} - \frac{31}{126} a^{5} - \frac{1}{42} a^{4} + \frac{17}{63} a^{3} - \frac{53}{126} a^{2} + \frac{5}{14}$, $\frac{1}{8623566} a^{15} + \frac{583}{1231938} a^{14} - \frac{28715}{8623566} a^{13} - \frac{33034}{1437261} a^{12} + \frac{270574}{4311783} a^{11} - \frac{123101}{8623566} a^{10} + \frac{21317}{410646} a^{9} - \frac{1981387}{8623566} a^{8} - \frac{3828767}{8623566} a^{7} + \frac{368903}{958174} a^{6} - \frac{3793567}{8623566} a^{5} + \frac{1248083}{4311783} a^{4} + \frac{2028694}{4311783} a^{3} - \frac{3965621}{8623566} a^{2} + \frac{642979}{2874522} a + \frac{390809}{958174}$

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

Class group and class number

Trivial group, which has 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:  \( -\frac{11886797}{4311783} a^{15} + \frac{84806537}{8623566} a^{14} - \frac{6393207}{958174} a^{13} - \frac{502861745}{8623566} a^{12} + \frac{34452076}{205323} a^{11} - \frac{130308364}{1437261} a^{10} - \frac{3421444583}{8623566} a^{9} + \frac{398611209}{479087} a^{8} - \frac{193039059}{479087} a^{7} - \frac{878670839}{1231938} a^{6} + \frac{394476281}{410646} a^{5} + \frac{795796721}{2874522} a^{4} - \frac{5812199018}{4311783} a^{3} + \frac{4669477187}{4311783} a^{2} - \frac{362789319}{958174} a + \frac{24143282}{479087} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8643.70735911 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2$ (as 16T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{-5})\), 8.0.40960000.1, 8.0.5898240000.3 x2, 8.0.235929600.1 x2

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 8 siblings: 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 R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$