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Properties

Label 16.0.97283799690...7489.1
Degree $16$
Signature $[0, 8]$
Discriminant $29^{14}\cdot 83^{6}$
Root discriminant $99.83$
Ramified primes $29, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28188139, -20847139, 10205840, -2392475, 2452312, -1027637, 40086, -207338, 134918, -3768, 1215, -4548, 920, -88, 55, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 55*x^14 - 88*x^13 + 920*x^12 - 4548*x^11 + 1215*x^10 - 3768*x^9 + 134918*x^8 - 207338*x^7 + 40086*x^6 - 1027637*x^5 + 2452312*x^4 - 2392475*x^3 + 10205840*x^2 - 20847139*x + 28188139)
 
gp: K = bnfinit(x^16 - 4*x^15 + 55*x^14 - 88*x^13 + 920*x^12 - 4548*x^11 + 1215*x^10 - 3768*x^9 + 134918*x^8 - 207338*x^7 + 40086*x^6 - 1027637*x^5 + 2452312*x^4 - 2392475*x^3 + 10205840*x^2 - 20847139*x + 28188139, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 55 x^{14} - 88 x^{13} + 920 x^{12} - 4548 x^{11} + 1215 x^{10} - 3768 x^{9} + 134918 x^{8} - 207338 x^{7} + 40086 x^{6} - 1027637 x^{5} + 2452312 x^{4} - 2392475 x^{3} + 10205840 x^{2} - 20847139 x + 28188139 \)

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:  \(97283799690045652521048020437489=29^{14}\cdot 83^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.83$
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:  $29, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{245} a^{11} - \frac{2}{245} a^{10} + \frac{12}{245} a^{9} + \frac{2}{245} a^{8} - \frac{6}{245} a^{7} - \frac{9}{35} a^{6} + \frac{13}{245} a^{5} + \frac{79}{245} a^{4} - \frac{1}{49} a^{3} + \frac{61}{245} a^{2} - \frac{64}{245} a - \frac{11}{35}$, $\frac{1}{245} a^{12} + \frac{8}{245} a^{10} - \frac{9}{245} a^{9} - \frac{2}{245} a^{8} - \frac{1}{49} a^{7} - \frac{113}{245} a^{6} + \frac{3}{7} a^{5} - \frac{92}{245} a^{4} + \frac{86}{245} a^{3} + \frac{58}{245} a^{2} - \frac{6}{49} a + \frac{13}{35}$, $\frac{1}{3185} a^{13} + \frac{6}{3185} a^{12} + \frac{2}{3185} a^{11} + \frac{12}{245} a^{10} + \frac{152}{3185} a^{9} + \frac{76}{3185} a^{8} - \frac{212}{3185} a^{7} + \frac{304}{637} a^{6} + \frac{43}{637} a^{5} - \frac{160}{637} a^{4} + \frac{324}{3185} a^{3} + \frac{582}{3185} a^{2} - \frac{18}{637} a - \frac{171}{455}$, $\frac{1}{131384435} a^{14} + \frac{9211}{131384435} a^{13} - \frac{150844}{131384435} a^{12} + \frac{172239}{131384435} a^{11} - \frac{4075647}{131384435} a^{10} + \frac{6381096}{131384435} a^{9} - \frac{2103289}{131384435} a^{8} - \frac{6441193}{131384435} a^{7} - \frac{62771001}{131384435} a^{6} - \frac{2065682}{10106495} a^{5} - \frac{55792602}{131384435} a^{4} - \frac{59151499}{131384435} a^{3} + \frac{9519546}{26276887} a^{2} + \frac{473042}{10106495} a + \frac{5615829}{18769205}$, $\frac{1}{20448922968869517595199010637405431925} a^{15} + \frac{10431343165933831933651908903}{20448922968869517595199010637405431925} a^{14} + \frac{294157137975962497598384175065091}{20448922968869517595199010637405431925} a^{13} - \frac{37949581159134201800341150342453056}{20448922968869517595199010637405431925} a^{12} + \frac{21102201688434226286134557542105968}{20448922968869517595199010637405431925} a^{11} + \frac{41839450871213596218687104609332273}{20448922968869517595199010637405431925} a^{10} + \frac{19506694668083695949795429997711222}{1572994074528424430399923895185033225} a^{9} - \frac{1305464286031305505447934738782211171}{20448922968869517595199010637405431925} a^{8} - \frac{859232986956127385906758783277110524}{20448922968869517595199010637405431925} a^{7} + \frac{1702630237899713925589769592661926964}{20448922968869517595199010637405431925} a^{6} - \frac{10188972033645724121869538545182624591}{20448922968869517595199010637405431925} a^{5} + \frac{9286746677496090166609405338604430161}{20448922968869517595199010637405431925} a^{4} + \frac{5217558548449692587383983467188480594}{20448922968869517595199010637405431925} a^{3} - \frac{262768263332085473654407933757554341}{2921274709838502513599858662486490275} a^{2} - \frac{6172275119833646476209064886721114974}{20448922968869517595199010637405431925} a + \frac{213740286936683713583349639425923469}{2921274709838502513599858662486490275}$

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:  $7$
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:  \( 23434432274.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

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 $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.2.69803.1, 4.0.24389.1, 4.2.2024287.1, 8.0.118834397892701.1 x2, 8.0.4097737858369.1

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
29Data not computed
83Data not computed