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Properties

Label 16.8.83680629612...2769.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 89^{6}$
Root discriminant $64.22$
Ramified primes $17, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2299691, 10197497, -13000438, 3176965, 1781862, -283428, 704747, -120622, -13152, 11597, -22188, -2446, -143, -146, 44, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 44*x^14 - 146*x^13 - 143*x^12 - 2446*x^11 - 22188*x^10 + 11597*x^9 - 13152*x^8 - 120622*x^7 + 704747*x^6 - 283428*x^5 + 1781862*x^4 + 3176965*x^3 - 13000438*x^2 + 10197497*x - 2299691)
 
gp: K = bnfinit(x^16 - 2*x^15 + 44*x^14 - 146*x^13 - 143*x^12 - 2446*x^11 - 22188*x^10 + 11597*x^9 - 13152*x^8 - 120622*x^7 + 704747*x^6 - 283428*x^5 + 1781862*x^4 + 3176965*x^3 - 13000438*x^2 + 10197497*x - 2299691, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 44 x^{14} - 146 x^{13} - 143 x^{12} - 2446 x^{11} - 22188 x^{10} + 11597 x^{9} - 13152 x^{8} - 120622 x^{7} + 704747 x^{6} - 283428 x^{5} + 1781862 x^{4} + 3176965 x^{3} - 13000438 x^{2} + 10197497 x - 2299691 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(83680629612698426645202702769=17^{14}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.22$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{560114} a^{14} - \frac{22601}{280057} a^{13} + \frac{89811}{560114} a^{12} - \frac{116237}{560114} a^{11} + \frac{43601}{280057} a^{10} - \frac{107919}{560114} a^{9} + \frac{265103}{560114} a^{8} + \frac{11725}{280057} a^{7} - \frac{164617}{560114} a^{6} + \frac{269805}{560114} a^{5} + \frac{16944}{280057} a^{4} - \frac{246915}{560114} a^{3} - \frac{193257}{560114} a^{2} + \frac{32052}{280057} a + \frac{127207}{560114}$, $\frac{1}{115416946783557583875847812270893446736552400410542} a^{15} + \frac{5846282952552811846434861837048588056556459}{115416946783557583875847812270893446736552400410542} a^{14} - \frac{12100013917730123700670629108505951704867598566825}{57708473391778791937923906135446723368276200205271} a^{13} + \frac{401408463785508034553097253914771647030210029401}{57708473391778791937923906135446723368276200205271} a^{12} + \frac{8593923115323968370515998987402101415082097924167}{115416946783557583875847812270893446736552400410542} a^{11} + \frac{4732398249443981190757310472449559731541839666573}{57708473391778791937923906135446723368276200205271} a^{10} - \frac{7889687247639629891203562387138926450337648346361}{57708473391778791937923906135446723368276200205271} a^{9} + \frac{7500169481824640324885396987354403721575801958427}{115416946783557583875847812270893446736552400410542} a^{8} + \frac{13230106618233502518192304334283548167592018215384}{57708473391778791937923906135446723368276200205271} a^{7} + \frac{11844724203832755578196640442185451670731605446796}{57708473391778791937923906135446723368276200205271} a^{6} - \frac{34762897759647826561474822617773087163409902796349}{115416946783557583875847812270893446736552400410542} a^{5} + \frac{27072641105161564717095151975988126423035006805229}{57708473391778791937923906135446723368276200205271} a^{4} - \frac{13565356109391863956473576910449277764657793596146}{57708473391778791937923906135446723368276200205271} a^{3} - \frac{31303661816470154051665586271455350312450555563239}{115416946783557583875847812270893446736552400410542} a^{2} + \frac{9876488952850015211777681041761457383679550240415}{57708473391778791937923906135446723368276200205271} a + \frac{36384036901906265139547032841174080554610375288939}{115416946783557583875847812270893446736552400410542}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 304838967.255 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.289276043966137.1, \(\Q(\zeta_{17})^+\), 8.4.17016237880361.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$