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Properties

Label 16.0.66283426716...3249.9
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 89^{8}$
Root discriminant $112.55$
Ramified primes $17, 89$
Class number $40$ (GRH)
Class group $[2, 2, 10]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22248343, -75302539, 59480154, 4456149, 678072, 6468196, 2070325, -145806, 311548, -28397, 24526, -4118, 1571, -148, 40, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 40*x^14 - 148*x^13 + 1571*x^12 - 4118*x^11 + 24526*x^10 - 28397*x^9 + 311548*x^8 - 145806*x^7 + 2070325*x^6 + 6468196*x^5 + 678072*x^4 + 4456149*x^3 + 59480154*x^2 - 75302539*x + 22248343)
 
gp: K = bnfinit(x^16 - 4*x^15 + 40*x^14 - 148*x^13 + 1571*x^12 - 4118*x^11 + 24526*x^10 - 28397*x^9 + 311548*x^8 - 145806*x^7 + 2070325*x^6 + 6468196*x^5 + 678072*x^4 + 4456149*x^3 + 59480154*x^2 - 75302539*x + 22248343, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 40 x^{14} - 148 x^{13} + 1571 x^{12} - 4118 x^{11} + 24526 x^{10} - 28397 x^{9} + 311548 x^{8} - 145806 x^{7} + 2070325 x^{6} + 6468196 x^{5} + 678072 x^{4} + 4456149 x^{3} + 59480154 x^{2} - 75302539 x + 22248343 \)

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:  \(662834267162184237456650608633249=17^{14}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.55$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{26} a^{13} + \frac{5}{26} a^{12} + \frac{5}{13} a^{11} + \frac{9}{26} a^{10} + \frac{5}{26} a^{9} + \frac{5}{13} a^{8} + \frac{9}{26} a^{7} - \frac{11}{26} a^{6} - \frac{2}{13} a^{5} + \frac{11}{26} a^{4} - \frac{7}{26} a^{3} - \frac{5}{13} a^{2} + \frac{11}{26} a - \frac{1}{2}$, $\frac{1}{26} a^{14} - \frac{1}{13} a^{12} + \frac{11}{26} a^{11} + \frac{6}{13} a^{10} - \frac{1}{13} a^{9} + \frac{11}{26} a^{8} - \frac{2}{13} a^{7} + \frac{6}{13} a^{6} + \frac{5}{26} a^{5} - \frac{5}{13} a^{4} + \frac{6}{13} a^{3} + \frac{9}{26} a^{2} + \frac{5}{13} a$, $\frac{1}{222978046845111782890543009229574327633223868410816287442} a^{15} + \frac{1588529886236406297709805317678375283610293870139819413}{111489023422555891445271504614787163816611934205408143721} a^{14} - \frac{2759362668243844646908308833725313760898845538107743635}{222978046845111782890543009229574327633223868410816287442} a^{13} + \frac{25081797358632786037678199231763808069318163536729400195}{111489023422555891445271504614787163816611934205408143721} a^{12} + \frac{18464050045933411914884685739956134418804292211599458118}{111489023422555891445271504614787163816611934205408143721} a^{11} + \frac{61118105638516248767797142087989457641692808721303410197}{222978046845111782890543009229574327633223868410816287442} a^{10} - \frac{26386812844497297860002263708155204054113882396425503638}{111489023422555891445271504614787163816611934205408143721} a^{9} - \frac{54070712686605957309675089817439523935031852893491288006}{111489023422555891445271504614787163816611934205408143721} a^{8} + \frac{12080067933952593734105717339342443748824686466258459201}{222978046845111782890543009229574327633223868410816287442} a^{7} - \frac{24092360219035218536019034810815836719220613833400694647}{111489023422555891445271504614787163816611934205408143721} a^{6} + \frac{13246200973855561566646203975971539479808570107218538244}{111489023422555891445271504614787163816611934205408143721} a^{5} - \frac{62084214735187191544929920847574391273291579114428572931}{222978046845111782890543009229574327633223868410816287442} a^{4} + \frac{20057064984039764265915046104849766990754569294194914081}{111489023422555891445271504614787163816611934205408143721} a^{3} + \frac{55172494842303970866930406319044043915575262489834776670}{111489023422555891445271504614787163816611934205408143721} a^{2} - \frac{110761358253993065535077489733609018555547922011141713955}{222978046845111782890543009229574327633223868410816287442} a - \frac{170902444114095383503762425316328560116591436787991981}{364939520204765602112181684500121649154212550590534022}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}$, which has order $40$ (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:  \( 148444206.244 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.289276043966137.1, 8.0.3250292628833.2, 8.4.17016237880361.1

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

Sibling fields

Degree 16 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
89Data not computed