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Properties

Label 16.0.12551165043...4321.1
Degree $16$
Signature $[0, 8]$
Discriminant $29^{14}\cdot 59^{6}$
Root discriminant $87.84$
Ramified primes $29, 59$
Class number $3$ (GRH)
Class group $[3]$ (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![1841449, 0, 762575, 0, 223934, 0, -98607, 0, 36700, 0, -5833, 0, 484, 0, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 + 484*x^12 - 5833*x^10 + 36700*x^8 - 98607*x^6 + 223934*x^4 + 762575*x^2 + 1841449)
 
gp: K = bnfinit(x^16 - 24*x^14 + 484*x^12 - 5833*x^10 + 36700*x^8 - 98607*x^6 + 223934*x^4 + 762575*x^2 + 1841449, 1)
 

Normalized defining polynomial

\( x^{16} - 24 x^{14} + 484 x^{12} - 5833 x^{10} + 36700 x^{8} - 98607 x^{6} + 223934 x^{4} + 762575 x^{2} + 1841449 \)

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:  \(12551165043538064715930071464321=29^{14}\cdot 59^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.84$
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:  $29, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{10} - \frac{1}{2} a^{7} + \frac{2}{7} a^{6} - \frac{1}{2} a^{5} - \frac{3}{7} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{2}{7}$, $\frac{1}{14} a^{11} + \frac{2}{7} a^{7} - \frac{1}{2} a^{6} + \frac{1}{14} a^{5} - \frac{1}{2} a^{4} - \frac{3}{14} a - \frac{1}{2}$, $\frac{1}{14} a^{12} - \frac{3}{14} a^{8} - \frac{1}{2} a^{7} + \frac{1}{14} a^{6} - \frac{1}{2} a^{3} - \frac{3}{14} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{13} - \frac{3}{14} a^{9} + \frac{1}{14} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{2}{7} a^{3} - \frac{1}{2}$, $\frac{1}{594591989650175016206} a^{14} + \frac{2352927718624049523}{594591989650175016206} a^{12} + \frac{46288308532503799}{297295994825087508103} a^{10} + \frac{3503665503122901896}{42470856403583929729} a^{8} + \frac{115671439171379895469}{297295994825087508103} a^{6} + \frac{42381746697410211713}{297295994825087508103} a^{4} + \frac{105042745819779318218}{297295994825087508103} a^{2} - \frac{1}{2} a - \frac{1227500756218901193}{10077830333053813834}$, $\frac{1}{13675615761954025372738} a^{15} - \frac{189942389956815659019}{6837807880977012686369} a^{13} + \frac{24275673417552603245}{976829697282430383767} a^{11} + \frac{1638418201858049642974}{6837807880977012686369} a^{9} - \frac{2359379362275859922531}{13675615761954025372738} a^{7} - \frac{1}{2} a^{6} - \frac{406532228350352961125}{1953659394564860767534} a^{5} + \frac{232455315030531107405}{6837807880977012686369} a^{3} + \frac{13423227585786932958}{115895048830118859091} a - \frac{1}{2}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 176629196.777 \) (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

\(\Q(\sqrt{29}) \), 4.0.24389.1, 4.2.49619.1, 4.2.1438951.1, 8.0.60046819431629.1 x2, 8.0.2070579980401.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ R

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
59Data not computed