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Properties

Label 16.0.64470132040...4513.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 83^{8}$
Root discriminant $129.74$
Ramified primes $17, 83$
Class number $16$ (GRH)
Class group $[16]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23119168, 14558768, -8650820, -13379230, 10843313, -5687148, 3114838, -1143564, 415396, -96354, 29822, -6452, 1682, -308, 64, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 64*x^14 - 308*x^13 + 1682*x^12 - 6452*x^11 + 29822*x^10 - 96354*x^9 + 415396*x^8 - 1143564*x^7 + 3114838*x^6 - 5687148*x^5 + 10843313*x^4 - 13379230*x^3 - 8650820*x^2 + 14558768*x + 23119168)
 
gp: K = bnfinit(x^16 - 8*x^15 + 64*x^14 - 308*x^13 + 1682*x^12 - 6452*x^11 + 29822*x^10 - 96354*x^9 + 415396*x^8 - 1143564*x^7 + 3114838*x^6 - 5687148*x^5 + 10843313*x^4 - 13379230*x^3 - 8650820*x^2 + 14558768*x + 23119168, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 64 x^{14} - 308 x^{13} + 1682 x^{12} - 6452 x^{11} + 29822 x^{10} - 96354 x^{9} + 415396 x^{8} - 1143564 x^{7} + 3114838 x^{6} - 5687148 x^{5} + 10843313 x^{4} - 13379230 x^{3} - 8650820 x^{2} + 14558768 x + 23119168 \)

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:  \(6447013204011288145829796273674513=17^{15}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.74$
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:  $17, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{16} a^{6} + \frac{1}{8} a^{5} - \frac{3}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{10} + \frac{3}{16} a^{7} - \frac{3}{16} a^{4}$, $\frac{1}{126761616008275186842592} a^{14} - \frac{7}{126761616008275186842592} a^{13} + \frac{56155084026674137679}{7922601000517199177662} a^{12} - \frac{5390888066560717217093}{126761616008275186842592} a^{11} - \frac{1091819416724663267545}{9750893539098091295584} a^{10} - \frac{199277499303088540489}{3961300500258599588831} a^{9} + \frac{14633985113857772293195}{126761616008275186842592} a^{8} - \frac{16193621317268385365333}{126761616008275186842592} a^{7} + \frac{389639527012571857553}{15845202001034398355324} a^{6} + \frac{151109104230135238507}{1891964418033958012576} a^{5} - \frac{27177085343359867203}{9750893539098091295584} a^{4} + \frac{2748543454652923276257}{15845202001034398355324} a^{3} - \frac{6946368896913928457899}{15845202001034398355324} a^{2} + \frac{5914631846087010468311}{15845202001034398355324} a + \frac{1788118137051144712753}{3961300500258599588831}$, $\frac{1}{40975565613058945871681021408} a^{15} + \frac{161617}{40975565613058945871681021408} a^{14} - \frac{295857201222141814098485149}{20487782806529472935840510704} a^{13} + \frac{149900937715874514447649947}{40975565613058945871681021408} a^{12} + \frac{1233914141739205454167140171}{40975565613058945871681021408} a^{11} - \frac{559690486324742407878446967}{20487782806529472935840510704} a^{10} + \frac{3663213663639939518790820011}{40975565613058945871681021408} a^{9} - \frac{599672110800719093128111589}{40975565613058945871681021408} a^{8} - \frac{601929474185349765409643291}{20487782806529472935840510704} a^{7} - \frac{4255770861449689847670074575}{40975565613058945871681021408} a^{6} + \frac{1586482117962758630922386713}{40975565613058945871681021408} a^{5} - \frac{373803046198199438519130969}{20487782806529472935840510704} a^{4} + \frac{2323314068849596975830354291}{5121945701632368233960127676} a^{3} + \frac{398478593551968387539161333}{1280486425408092058490031919} a^{2} - \frac{17523703157907095002481167}{48320242468229888999623846} a - \frac{579258805666927196423520694}{1280486425408092058490031919}$

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

Class group and class number

$C_{16}$, which has order $16$ (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:  \( 13918762278.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-1411}) \), 4.0.33845657.1, 8.0.19473984461948033.1

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 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
$83$83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$