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Properties

Label 16.16.1650129926...2193.1
Degree $16$
Signature $[16, 0]$
Discriminant $7^{8}\cdot 17^{15}$
Root discriminant $37.68$
Ramified primes $7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-101, 4453, -4453, -21659, 21659, 24037, -24037, -11867, 11867, 3093, -3093, -443, 443, 33, -33, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 33*x^14 + 33*x^13 + 443*x^12 - 443*x^11 - 3093*x^10 + 3093*x^9 + 11867*x^8 - 11867*x^7 - 24037*x^6 + 24037*x^5 + 21659*x^4 - 21659*x^3 - 4453*x^2 + 4453*x - 101)
 
gp: K = bnfinit(x^16 - x^15 - 33*x^14 + 33*x^13 + 443*x^12 - 443*x^11 - 3093*x^10 + 3093*x^9 + 11867*x^8 - 11867*x^7 - 24037*x^6 + 24037*x^5 + 21659*x^4 - 21659*x^3 - 4453*x^2 + 4453*x - 101, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 33 x^{14} + 33 x^{13} + 443 x^{12} - 443 x^{11} - 3093 x^{10} + 3093 x^{9} + 11867 x^{8} - 11867 x^{7} - 24037 x^{6} + 24037 x^{5} + 21659 x^{4} - 21659 x^{3} - 4453 x^{2} + 4453 x - 101 \)

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:  \(16501299269766837593302193=7^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.68$
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:  $7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(119=7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{119}(64,·)$, $\chi_{119}(1,·)$, $\chi_{119}(6,·)$, $\chi_{119}(8,·)$, $\chi_{119}(15,·)$, $\chi_{119}(20,·)$, $\chi_{119}(90,·)$, $\chi_{119}(27,·)$, $\chi_{119}(97,·)$, $\chi_{119}(36,·)$, $\chi_{119}(41,·)$, $\chi_{119}(106,·)$, $\chi_{119}(43,·)$, $\chi_{119}(48,·)$, $\chi_{119}(50,·)$, $\chi_{119}(62,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{271} a^{9} - \frac{96}{271} a^{8} - \frac{18}{271} a^{7} - \frac{90}{271} a^{6} + \frac{108}{271} a^{5} - \frac{92}{271} a^{4} + \frac{31}{271} a^{3} + \frac{93}{271} a^{2} - \frac{127}{271} a - \frac{91}{271}$, $\frac{1}{271} a^{10} - \frac{20}{271} a^{8} + \frac{79}{271} a^{7} - \frac{131}{271} a^{6} - \frac{22}{271} a^{5} - \frac{129}{271} a^{4} + \frac{88}{271} a^{3} + \frac{129}{271} a^{2} - \frac{88}{271} a - \frac{64}{271}$, $\frac{1}{271} a^{11} + \frac{56}{271} a^{8} + \frac{51}{271} a^{7} + \frac{75}{271} a^{6} + \frac{134}{271} a^{5} - \frac{126}{271} a^{4} - \frac{64}{271} a^{3} - \frac{125}{271} a^{2} + \frac{106}{271} a + \frac{77}{271}$, $\frac{1}{271} a^{12} + \frac{7}{271} a^{8} - \frac{1}{271} a^{7} + \frac{25}{271} a^{6} + \frac{59}{271} a^{5} - \frac{61}{271} a^{4} + \frac{36}{271} a^{3} + \frac{47}{271} a^{2} - \frac{128}{271} a - \frac{53}{271}$, $\frac{1}{271} a^{13} + \frac{129}{271} a^{8} - \frac{120}{271} a^{7} - \frac{124}{271} a^{6} - \frac{4}{271} a^{5} - \frac{133}{271} a^{4} + \frac{101}{271} a^{3} + \frac{34}{271} a^{2} + \frac{23}{271} a + \frac{95}{271}$, $\frac{1}{271} a^{14} + \frac{69}{271} a^{8} + \frac{30}{271} a^{7} - \frac{47}{271} a^{6} + \frac{27}{271} a^{5} + \frac{45}{271} a^{4} + \frac{100}{271} a^{3} - \frac{50}{271} a^{2} - \frac{53}{271} a + \frac{86}{271}$, $\frac{1}{271} a^{15} - \frac{121}{271} a^{8} + \frac{111}{271} a^{7} + \frac{4}{271} a^{6} - \frac{90}{271} a^{5} - \frac{56}{271} a^{4} - \frac{21}{271} a^{3} + \frac{34}{271} a^{2} - \frac{94}{271} a + \frac{46}{271}$

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:  \( 18808281.0978 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ R $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
7Data not computed
17Data not computed