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Properties

Label 16.16.6577258849...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{8}\cdot 17^{14}$
Root discriminant $26.68$
Ramified primes $5, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -46, 0, 325, 74, -844, -189, 1014, 196, -601, -92, 172, 17, -22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 22*x^14 + 17*x^13 + 172*x^12 - 92*x^11 - 601*x^10 + 196*x^9 + 1014*x^8 - 189*x^7 - 844*x^6 + 74*x^5 + 325*x^4 - 46*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - x^15 - 22*x^14 + 17*x^13 + 172*x^12 - 92*x^11 - 601*x^10 + 196*x^9 + 1014*x^8 - 189*x^7 - 844*x^6 + 74*x^5 + 325*x^4 - 46*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 22 x^{14} + 17 x^{13} + 172 x^{12} - 92 x^{11} - 601 x^{10} + 196 x^{9} + 1014 x^{8} - 189 x^{7} - 844 x^{6} + 74 x^{5} + 325 x^{4} - 46 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(65772588499765987890625=5^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.68$
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:  $5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(85=5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{85}(64,·)$, $\chi_{85}(1,·)$, $\chi_{85}(66,·)$, $\chi_{85}(4,·)$, $\chi_{85}(69,·)$, $\chi_{85}(9,·)$, $\chi_{85}(76,·)$, $\chi_{85}(16,·)$, $\chi_{85}(81,·)$, $\chi_{85}(19,·)$, $\chi_{85}(84,·)$, $\chi_{85}(21,·)$, $\chi_{85}(26,·)$, $\chi_{85}(36,·)$, $\chi_{85}(49,·)$, $\chi_{85}(59,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{183345026} a^{15} - \frac{720663}{183345026} a^{14} + \frac{29652739}{183345026} a^{13} - \frac{6032797}{183345026} a^{12} + \frac{38622761}{183345026} a^{11} - \frac{72773275}{183345026} a^{10} + \frac{5945279}{183345026} a^{9} - \frac{38414421}{183345026} a^{8} + \frac{89629411}{183345026} a^{7} - \frac{57930471}{183345026} a^{6} - \frac{15036833}{183345026} a^{5} - \frac{41945697}{183345026} a^{4} + \frac{25420041}{183345026} a^{3} - \frac{64296813}{183345026} a^{2} - \frac{60208255}{183345026} a - \frac{41176638}{91672513}$

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:  $15$
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:  \( 1299965.95967 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.4913.1, 4.4.122825.1, 8.8.15085980625.1, \(\Q(\zeta_{17})^+\), 8.8.256461670625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
17Data not computed