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Properties

Label 16.4.59734284026...5129.1
Degree $16$
Signature $[4, 6]$
Discriminant $13^{14}\cdot 79^{8}$
Root discriminant $83.85$
Ramified primes $13, 79$
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![-754857, -3048039, -4453919, -2949122, -814270, 71270, 14598, -96551, -50858, -17693, -1977, -2071, 102, -71, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 24*x^14 - 71*x^13 + 102*x^12 - 2071*x^11 - 1977*x^10 - 17693*x^9 - 50858*x^8 - 96551*x^7 + 14598*x^6 + 71270*x^5 - 814270*x^4 - 2949122*x^3 - 4453919*x^2 - 3048039*x - 754857)
 
gp: K = bnfinit(x^16 - 6*x^15 + 24*x^14 - 71*x^13 + 102*x^12 - 2071*x^11 - 1977*x^10 - 17693*x^9 - 50858*x^8 - 96551*x^7 + 14598*x^6 + 71270*x^5 - 814270*x^4 - 2949122*x^3 - 4453919*x^2 - 3048039*x - 754857, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 24 x^{14} - 71 x^{13} + 102 x^{12} - 2071 x^{11} - 1977 x^{10} - 17693 x^{9} - 50858 x^{8} - 96551 x^{7} + 14598 x^{6} + 71270 x^{5} - 814270 x^{4} - 2949122 x^{3} - 4453919 x^{2} - 3048039 x - 754857 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5973428402662444840532734135129=13^{14}\cdot 79^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.85$
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:  $13, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{3} a^{4} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{18} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{18} a^{6} + \frac{1}{3} a^{4} + \frac{7}{18} a^{3} + \frac{7}{18} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{162} a^{13} + \frac{1}{81} a^{12} - \frac{10}{81} a^{10} + \frac{5}{54} a^{9} - \frac{17}{162} a^{8} + \frac{13}{162} a^{7} + \frac{7}{81} a^{6} - \frac{10}{27} a^{5} - \frac{11}{162} a^{4} - \frac{7}{54} a^{3} + \frac{35}{81} a^{2} + \frac{1}{18} a + \frac{2}{9}$, $\frac{1}{217566} a^{14} - \frac{457}{217566} a^{13} + \frac{275}{24174} a^{12} - \frac{179}{6399} a^{11} - \frac{253}{72522} a^{10} - \frac{626}{6399} a^{9} + \frac{11749}{217566} a^{8} + \frac{16279}{108783} a^{7} + \frac{3067}{72522} a^{6} - \frac{32519}{217566} a^{5} - \frac{1619}{36261} a^{4} + \frac{48752}{108783} a^{3} - \frac{5485}{12087} a^{2} - \frac{6131}{24174} a + \frac{159}{2686}$, $\frac{1}{16095697248173322099461489468139196998} a^{15} + \frac{3311405835096847060810657549973}{2682616208028887016576914911356532833} a^{14} + \frac{13809124601218333909172116202641699}{5365232416057774033153829822713065666} a^{13} + \frac{638326019891117805062888511416209}{105200635608975961434388820053197366} a^{12} - \frac{239881605626192164683198483497057843}{16095697248173322099461489468139196998} a^{11} + \frac{101079183488279608058466300550971565}{946805720480783652909499380478776294} a^{10} + \frac{948228727874437161463594315446462061}{8047848624086661049730744734069598499} a^{9} + \frac{493923921376831728967090620700270655}{8047848624086661049730744734069598499} a^{8} - \frac{7414931570423833450904102981950453}{2682616208028887016576914911356532833} a^{7} + \frac{8989148183384285069516926998356506}{2682616208028887016576914911356532833} a^{6} + \frac{7599821246151073284340515207086735875}{16095697248173322099461489468139196998} a^{5} + \frac{3329936381722271418973975862792919359}{16095697248173322099461489468139196998} a^{4} - \frac{8032519881262729472438586137669232611}{16095697248173322099461489468139196998} a^{3} + \frac{6201022439492478962756520421372201051}{16095697248173322099461489468139196998} a^{2} + \frac{759245494963719031940545856377147295}{1788410805352591344384609940904355222} a + \frac{45666611349566946170909922329438273}{105200635608975961434388820053197366}$

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:  $9$
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:  \( 2293989833.65 \) (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

\(\Q(\sqrt{13}) \), 4.2.173563.1, 4.4.13711477.1, 4.2.13351.1, 8.4.2444059819779877.1 x2, 8.4.188004601521529.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed
$79$79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$