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Properties

Label 16.0.18102079666...7137.3
Degree $16$
Signature $[0, 8]$
Discriminant $47^{8}\cdot 97^{9}$
Root discriminant $89.87$
Ramified primes $47, 97$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63451136, -123625248, 94305600, -20454968, -15131120, 10502938, -1306251, -671891, 167258, 24342, -2787, -4442, 965, 28, -2, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 2*x^14 + 28*x^13 + 965*x^12 - 4442*x^11 - 2787*x^10 + 24342*x^9 + 167258*x^8 - 671891*x^7 - 1306251*x^6 + 10502938*x^5 - 15131120*x^4 - 20454968*x^3 + 94305600*x^2 - 123625248*x + 63451136)
 
gp: K = bnfinit(x^16 - 7*x^15 - 2*x^14 + 28*x^13 + 965*x^12 - 4442*x^11 - 2787*x^10 + 24342*x^9 + 167258*x^8 - 671891*x^7 - 1306251*x^6 + 10502938*x^5 - 15131120*x^4 - 20454968*x^3 + 94305600*x^2 - 123625248*x + 63451136, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 2 x^{14} + 28 x^{13} + 965 x^{12} - 4442 x^{11} - 2787 x^{10} + 24342 x^{9} + 167258 x^{8} - 671891 x^{7} - 1306251 x^{6} + 10502938 x^{5} - 15131120 x^{4} - 20454968 x^{3} + 94305600 x^{2} - 123625248 x + 63451136 \)

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:  \(18102079666797893193942794567137=47^{8}\cdot 97^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.87$
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:  $47, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{3}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{3}{16} a^{6} + \frac{1}{4} a^{5} - \frac{3}{16} a^{4} + \frac{3}{8} a^{3}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} + \frac{1}{16} a^{7} + \frac{5}{16} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{224} a^{13} + \frac{1}{224} a^{12} - \frac{5}{224} a^{11} - \frac{11}{224} a^{10} - \frac{1}{28} a^{9} - \frac{3}{32} a^{8} - \frac{31}{224} a^{7} - \frac{19}{224} a^{6} + \frac{13}{32} a^{5} + \frac{29}{112} a^{4} - \frac{1}{7} a^{3} - \frac{1}{4} a^{2} + \frac{2}{7} a$, $\frac{1}{896} a^{14} - \frac{1}{896} a^{13} + \frac{1}{128} a^{12} - \frac{1}{896} a^{11} - \frac{33}{896} a^{9} + \frac{95}{896} a^{8} + \frac{113}{896} a^{7} + \frac{185}{896} a^{6} + \frac{141}{448} a^{5} - \frac{9}{28} a^{4} - \frac{41}{112} a^{3} - \frac{3}{7} a^{2} + \frac{3}{28} a$, $\frac{1}{46735897282731501892895119541885158172672} a^{15} + \frac{42455201600494559264451565459401995}{1669139188668982210460539983638755649024} a^{14} - \frac{31450272459283142527218486104545982787}{23367948641365750946447559770942579086336} a^{13} + \frac{444209458984414682137205374267714805469}{23367948641365750946447559770942579086336} a^{12} - \frac{134129812590602703143506617435857093499}{6676556754675928841842159934555022596096} a^{11} + \frac{356399016942857073303292012762446122577}{6676556754675928841842159934555022596096} a^{10} - \frac{1446614059555441145477474164741839990491}{23367948641365750946447559770942579086336} a^{9} + \frac{697621473750719714819813789934453176649}{11683974320682875473223779885471289543168} a^{8} + \frac{4415054790158290422495064424421745144211}{23367948641365750946447559770942579086336} a^{7} - \frac{10105814516103107744235875728942822986961}{46735897282731501892895119541885158172672} a^{6} - \frac{10422123495678242719873777665349653297243}{23367948641365750946447559770942579086336} a^{5} - \frac{1219149453427082116139175200786108438827}{5841987160341437736611889942735644771584} a^{4} + \frac{43231730226910795528009531941853579855}{834569594334491105230269991819377824512} a^{3} - \frac{529456329994869220814429008790302290141}{1460496790085359434152972485683911192896} a^{2} - \frac{495509309322384883050258549997523163685}{1460496790085359434152972485683911192896} a - \frac{665489865287632481978771517517336572}{3260037477869105879805742155544444627}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( 33884343603.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-47}) \), 4.0.214273.1, 8.0.4453553097313.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$47$47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.7.4$x^{8} - 1515625$$8$$1$$7$$C_8$$[\ ]_{8}$