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Properties

Label 16.0.23518009076...8601.1
Degree $16$
Signature $[0, 8]$
Discriminant $47^{8}\cdot 61^{14}$
Root discriminant $250.16$
Ramified primes $47, 61$
Class number $30$ (GRH)
Class group $[30]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1426950625, 0, 2526369250, 0, 642125261, 0, 71800669, 0, 2952056, 0, -70442, 0, -4199, 0, 54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 54*x^14 - 4199*x^12 - 70442*x^10 + 2952056*x^8 + 71800669*x^6 + 642125261*x^4 + 2526369250*x^2 + 1426950625)
 
gp: K = bnfinit(x^16 + 54*x^14 - 4199*x^12 - 70442*x^10 + 2952056*x^8 + 71800669*x^6 + 642125261*x^4 + 2526369250*x^2 + 1426950625, 1)
 

Normalized defining polynomial

\( x^{16} + 54 x^{14} - 4199 x^{12} - 70442 x^{10} + 2952056 x^{8} + 71800669 x^{6} + 642125261 x^{4} + 2526369250 x^{2} + 1426950625 \)

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:  \(235180090762073460609416758748830068601=47^{8}\cdot 61^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $250.16$
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:  $47, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{10} a^{11} + \frac{1}{10} a^{7} - \frac{1}{2} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{10} a$, $\frac{1}{50} a^{12} + \frac{1}{50} a^{8} - \frac{1}{2} a^{7} - \frac{21}{50} a^{6} - \frac{1}{2} a^{5} - \frac{1}{5} a^{4} + \frac{9}{50} a^{2}$, $\frac{1}{250} a^{13} - \frac{49}{250} a^{9} + \frac{79}{250} a^{7} - \frac{6}{25} a^{5} - \frac{58}{125} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{48145432173454973707493744401750} a^{14} - \frac{47152860329090684462321892933}{9629086434690994741498748880350} a^{12} - \frac{2763116013626873910312463839462}{24072716086727486853746872200875} a^{10} - \frac{1018075157488432467853654544961}{48145432173454973707493744401750} a^{8} - \frac{1}{2} a^{7} + \frac{941916050557681882495423410531}{9629086434690994741498748880350} a^{6} - \frac{19580892689323881613004281390341}{48145432173454973707493744401750} a^{4} - \frac{1}{2} a^{3} - \frac{63449442271328131237222028421}{4814543217345497370749374440175} a^{2} - \frac{1}{2} a - \frac{62746959987596562022397528495}{192581728693819894829974977607}$, $\frac{1}{72747748014090465272023047791044250} a^{15} + \frac{4959972085710226581117027524849}{14549549602818093054404609558208850} a^{13} + \frac{490371719359332481366560639659101}{72747748014090465272023047791044250} a^{11} - \frac{7992169324050535301940813460518963}{36373874007045232636011523895522125} a^{9} - \frac{192795303015425132316459404062111}{14549549602818093054404609558208850} a^{7} + \frac{194827640392521697450346273012267}{36373874007045232636011523895522125} a^{5} - \frac{1}{2} a^{4} + \frac{3426350667478172240683677579555323}{7274774801409046527202304779104425} a^{3} + \frac{73059903832407397120108478825792}{1454954960281809305440460955820885} a - \frac{1}{2}$

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

Class group and class number

$C_{30}$, which has order $30$ (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:  \( 110204009551 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

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

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{-2867}) \), 4.4.501401029.1, 4.0.226981.1, \(\Q(\sqrt{-47}, \sqrt{61})\), 8.4.15335582504817789301.1 x2, 8.0.6942318924770389.1 x2, 8.0.251402991882258841.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$