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Properties

Label 16.0.76785869193...3616.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 17^{8}$
Root discriminant $20.20$
Ramified primes $2, 3, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![289, -1734, 5202, -8262, 7785, -3762, 540, 444, -196, 126, 18, -54, 9, -6, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^13 + 9*x^12 - 54*x^11 + 18*x^10 + 126*x^9 - 196*x^8 + 444*x^7 + 540*x^6 - 3762*x^5 + 7785*x^4 - 8262*x^3 + 5202*x^2 - 1734*x + 289)
 
gp: K = bnfinit(x^16 - 6*x^13 + 9*x^12 - 54*x^11 + 18*x^10 + 126*x^9 - 196*x^8 + 444*x^7 + 540*x^6 - 3762*x^5 + 7785*x^4 - 8262*x^3 + 5202*x^2 - 1734*x + 289, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{13} + 9 x^{12} - 54 x^{11} + 18 x^{10} + 126 x^{9} - 196 x^{8} + 444 x^{7} + 540 x^{6} - 3762 x^{5} + 7785 x^{4} - 8262 x^{3} + 5202 x^{2} - 1734 x + 289 \)

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:  \(767858691933644783616=2^{24}\cdot 3^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.20$
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 Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{680} a^{12} + \frac{1}{17} a^{11} + \frac{11}{170} a^{10} - \frac{7}{20} a^{9} - \frac{83}{340} a^{8} + \frac{53}{340} a^{7} + \frac{3}{340} a^{6} + \frac{4}{17} a^{5} + \frac{151}{340} a^{4} - \frac{1}{20} a^{3} + \frac{101}{340} a^{2} + \frac{7}{20} a + \frac{7}{40}$, $\frac{1}{680} a^{13} + \frac{18}{85} a^{11} + \frac{21}{340} a^{10} + \frac{87}{340} a^{9} - \frac{27}{340} a^{8} + \frac{93}{340} a^{7} - \frac{2}{17} a^{6} - \frac{159}{340} a^{5} + \frac{63}{340} a^{4} - \frac{69}{340} a^{3} + \frac{159}{340} a^{2} - \frac{13}{40} a$, $\frac{1}{2955280} a^{14} - \frac{251}{1477640} a^{13} + \frac{2141}{2955280} a^{12} - \frac{304223}{1477640} a^{11} - \frac{154371}{1477640} a^{10} - \frac{72993}{184705} a^{9} - \frac{364287}{738820} a^{8} + \frac{138099}{295528} a^{7} + \frac{801}{738820} a^{6} - \frac{457699}{1477640} a^{5} - \frac{162999}{738820} a^{4} - \frac{17893}{369410} a^{3} + \frac{266997}{2955280} a^{2} - \frac{19449}{43460} a - \frac{43121}{173840}$, $\frac{1}{222407998261040} a^{15} - \frac{8826343}{111203999130520} a^{14} - \frac{140173058979}{222407998261040} a^{13} + \frac{1776064101}{11120399913052} a^{12} - \frac{4476018418359}{22240799826104} a^{11} - \frac{1574672946451}{13900499891315} a^{10} - \frac{1427650182401}{27800999782630} a^{9} + \frac{15868262300277}{111203999130520} a^{8} + \frac{313742421661}{1635352928390} a^{7} - \frac{41841710997017}{111203999130520} a^{6} - \frac{36458834031}{271229266172} a^{5} + \frac{25417026551503}{55601999565260} a^{4} + \frac{1662991520749}{4196377325680} a^{3} - \frac{6687825067099}{13900499891315} a^{2} + \frac{2146045143707}{13082823427120} a + \frac{983560374347}{6541411713560}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( \frac{347309919}{102350666480} a^{15} + \frac{9414849}{6020627440} a^{14} + \frac{111029739}{102350666480} a^{13} - \frac{401826141}{20470133296} a^{12} + \frac{553891071}{25587666620} a^{11} - \frac{8971199479}{51175333240} a^{10} - \frac{450186789}{25587666620} a^{9} + \frac{20461719303}{51175333240} a^{8} - \frac{25135776777}{51175333240} a^{7} + \frac{13379726255}{10235066648} a^{6} + \frac{122292128631}{51175333240} a^{5} - \frac{294949776771}{25587666620} a^{4} + \frac{2194838163531}{102350666480} a^{3} - \frac{1972083124189}{102350666480} a^{2} + \frac{66948900831}{6020627440} a - \frac{3020353569}{1204125488} \) (order $4$)
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:  \( 12529.3021665 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-17}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{17})\), 4.2.1156.1 x2, 4.0.272.1 x2, 8.0.21381376.2, 8.2.6927565824.1 x4, 8.0.1630015488.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$