↑

Properties

Label 16.16.1531971467...6433.1
Degree $16$
Signature $[16, 0]$
Discriminant $3^{8}\cdot 13^{8}\cdot 17^{15}$
Root discriminant $88.94$
Ramified primes $3, 13, 17$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_{16}$ (as 16T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![270719731, 1429280269, -1429280269, -610719731, 610719731, 103280269, -103280269, -8919731, 8919731, 430269, -430269, -11731, 11731, 169, -169, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 169*x^14 + 169*x^13 + 11731*x^12 - 11731*x^11 - 430269*x^10 + 430269*x^9 + 8919731*x^8 - 8919731*x^7 - 103280269*x^6 + 103280269*x^5 + 610719731*x^4 - 610719731*x^3 - 1429280269*x^2 + 1429280269*x + 270719731)
 
gp: K = bnfinit(x^16 - x^15 - 169*x^14 + 169*x^13 + 11731*x^12 - 11731*x^11 - 430269*x^10 + 430269*x^9 + 8919731*x^8 - 8919731*x^7 - 103280269*x^6 + 103280269*x^5 + 610719731*x^4 - 610719731*x^3 - 1429280269*x^2 + 1429280269*x + 270719731, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 169 x^{14} + 169 x^{13} + 11731 x^{12} - 11731 x^{11} - 430269 x^{10} + 430269 x^{9} + 8919731 x^{8} - 8919731 x^{7} - 103280269 x^{6} + 103280269 x^{5} + 610719731 x^{4} - 610719731 x^{3} - 1429280269 x^{2} + 1429280269 x + 270719731 \)

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:  \(15319714679094816592806464576433=3^{8}\cdot 13^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.94$
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:  $3, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(663=3\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{663}(1,·)$, $\chi_{663}(194,·)$, $\chi_{663}(196,·)$, $\chi_{663}(584,·)$, $\chi_{663}(586,·)$, $\chi_{663}(274,·)$, $\chi_{663}(157,·)$, $\chi_{663}(350,·)$, $\chi_{663}(233,·)$, $\chi_{663}(428,·)$, $\chi_{663}(623,·)$, $\chi_{663}(625,·)$, $\chi_{663}(116,·)$, $\chi_{663}(118,·)$, $\chi_{663}(311,·)$, $\chi_{663}(508,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{91527049} a^{9} + \frac{33451207}{91527049} a^{8} - \frac{90}{91527049} a^{7} - \frac{21812139}{91527049} a^{6} + \frac{2700}{91527049} a^{5} - \frac{3858819}{91527049} a^{4} - \frac{30000}{91527049} a^{3} + \frac{30870552}{91527049} a^{2} + \frac{90000}{91527049} a - \frac{38588190}{91527049}$, $\frac{1}{91527049} a^{10} - \frac{100}{91527049} a^{8} - \frac{31596126}{91527049} a^{7} + \frac{3500}{91527049} a^{6} + \frac{15079644}{91527049} a^{5} - \frac{50000}{91527049} a^{4} - \frac{27011733}{91527049} a^{3} + \frac{250000}{91527049} a^{2} + \frac{43531616}{91527049} a - \frac{200000}{91527049}$, $\frac{1}{91527049} a^{11} + \frac{18550810}{91527049} a^{8} - \frac{5500}{91527049} a^{7} + \frac{30514920}{91527049} a^{6} + \frac{220000}{91527049} a^{5} + \frac{44741612}{91527049} a^{4} - \frac{2750000}{91527049} a^{3} + \frac{18667150}{91527049} a^{2} + \frac{8800000}{91527049} a - \frac{14682942}{91527049}$, $\frac{1}{91527049} a^{12} - \frac{6600}{91527049} a^{8} - \frac{38926111}{91527049} a^{7} + \frac{308000}{91527049} a^{6} + \frac{22850415}{91527049} a^{5} - \frac{4950000}{91527049} a^{4} - \frac{33017819}{91527049} a^{3} + \frac{26400000}{91527049} a^{2} - \frac{42682133}{91527049} a - \frac{22000000}{91527049}$, $\frac{1}{91527049} a^{13} - \frac{24202099}{91527049} a^{8} - \frac{286000}{91527049} a^{7} + \frac{34781092}{91527049} a^{6} + \frac{12870000}{91527049} a^{5} + \frac{34823452}{91527049} a^{4} + \frac{11454098}{91527049} a^{3} - \frac{36250007}{91527049} a^{2} + \frac{22837706}{91527049} a + \frac{37723367}{91527049}$, $\frac{1}{91527049} a^{14} - \frac{364000}{91527049} a^{8} - \frac{38285691}{91527049} a^{7} + \frac{19110000}{91527049} a^{6} + \frac{30177766}{91527049} a^{5} + \frac{38508196}{91527049} a^{4} - \frac{15140290}{91527049} a^{3} - \frac{10540980}{91527049} a^{2} - \frac{25605784}{91527049} a - \frac{4040167}{91527049}$, $\frac{1}{91527049} a^{15} - \frac{8374357}{91527049} a^{8} - \frac{13650000}{91527049} a^{7} + \frac{16974320}{91527049} a^{6} + \frac{14510657}{91527049} a^{5} + \frac{40364713}{91527049} a^{4} - \frac{38822149}{91527049} a^{3} - \frac{12010563}{91527049} a^{2} - \frac{10723709}{91527049} a + \frac{5887736}{91527049}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 6254168601.885075 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R $16$ $16$ $16$ R 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
3Data not computed
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17Data not computed