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Properties

Label 16.0.40986424795...6096.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 13^{12}$
Root discriminant $46.06$
Ramified primes $2, 13$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6241, -6952, 21012, -3988, -8980, 3472, 998, -44, 1364, -108, 220, -52, -39, 44, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 2*x^14 + 44*x^13 - 39*x^12 - 52*x^11 + 220*x^10 - 108*x^9 + 1364*x^8 - 44*x^7 + 998*x^6 + 3472*x^5 - 8980*x^4 - 3988*x^3 + 21012*x^2 - 6952*x + 6241)
 
gp: K = bnfinit(x^16 - 4*x^15 - 2*x^14 + 44*x^13 - 39*x^12 - 52*x^11 + 220*x^10 - 108*x^9 + 1364*x^8 - 44*x^7 + 998*x^6 + 3472*x^5 - 8980*x^4 - 3988*x^3 + 21012*x^2 - 6952*x + 6241, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 2 x^{14} + 44 x^{13} - 39 x^{12} - 52 x^{11} + 220 x^{10} - 108 x^{9} + 1364 x^{8} - 44 x^{7} + 998 x^{6} + 3472 x^{5} - 8980 x^{4} - 3988 x^{3} + 21012 x^{2} - 6952 x + 6241 \)

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:  \(409864247953326282266116096=2^{44}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.06$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(208=2^{4}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{208}(1,·)$, $\chi_{208}(5,·)$, $\chi_{208}(129,·)$, $\chi_{208}(73,·)$, $\chi_{208}(77,·)$, $\chi_{208}(21,·)$, $\chi_{208}(25,·)$, $\chi_{208}(157,·)$, $\chi_{208}(161,·)$, $\chi_{208}(105,·)$, $\chi_{208}(109,·)$, $\chi_{208}(177,·)$, $\chi_{208}(53,·)$, $\chi_{208}(57,·)$, $\chi_{208}(125,·)$, $\chi_{208}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{116590092603} a^{14} - \frac{2942005676}{38863364201} a^{13} - \frac{12050527582}{116590092603} a^{12} - \frac{105439292}{1689711487} a^{11} + \frac{4701331771}{38863364201} a^{10} - \frac{19325554351}{38863364201} a^{9} - \frac{11335762243}{38863364201} a^{8} + \frac{2750631092}{38863364201} a^{7} - \frac{32887332163}{116590092603} a^{6} - \frac{6271113768}{38863364201} a^{5} - \frac{26121332429}{116590092603} a^{4} + \frac{11034421382}{38863364201} a^{3} - \frac{45377800783}{116590092603} a^{2} + \frac{7348107993}{38863364201} a + \frac{37955525}{491941319}$, $\frac{1}{165413781954274185555603} a^{15} + \frac{676926295402}{165413781954274185555603} a^{14} + \frac{5955115598849711196167}{55137927318091395185201} a^{13} - \frac{1828466738440675101605}{165413781954274185555603} a^{12} + \frac{26819443938903999674609}{165413781954274185555603} a^{11} - \frac{63504620482181968304708}{165413781954274185555603} a^{10} + \frac{15315855998798538672362}{55137927318091395185201} a^{9} + \frac{3095964909574400515545}{55137927318091395185201} a^{8} - \frac{18053813320897538750905}{165413781954274185555603} a^{7} + \frac{20177538201057263373713}{165413781954274185555603} a^{6} + \frac{21673356399822038739171}{55137927318091395185201} a^{5} + \frac{10410935866644490916729}{165413781954274185555603} a^{4} - \frac{19591173350664518986042}{55137927318091395185201} a^{3} - \frac{68827016367928321230314}{165413781954274185555603} a^{2} + \frac{5045295745342120295402}{165413781954274185555603} a - \frac{659531308776530459459}{2093845341193344120957}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( 108889.88555195347 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{26}) \), \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{13})\), 4.4.346112.1, 4.0.2197.1, 4.0.140608.2, 4.0.4499456.2, 4.0.4499456.1, 8.8.119793516544.1, 8.0.19770609664.2, 8.0.20245104295936.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
13Data not computed