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Properties

Label 16.0.34336147906...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $19.21$
Ramified primes $3, 5, 11$
Class number $4$
Class group $[4]$
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, -2187, 2187, -1944, 648, 189, 54, 168, -137, 56, 6, 7, 8, -8, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 3*x^14 - 8*x^13 + 8*x^12 + 7*x^11 + 6*x^10 + 56*x^9 - 137*x^8 + 168*x^7 + 54*x^6 + 189*x^5 + 648*x^4 - 1944*x^3 + 2187*x^2 - 2187*x + 6561)
 
gp: K = bnfinit(x^16 - x^15 + 3*x^14 - 8*x^13 + 8*x^12 + 7*x^11 + 6*x^10 + 56*x^9 - 137*x^8 + 168*x^7 + 54*x^6 + 189*x^5 + 648*x^4 - 1944*x^3 + 2187*x^2 - 2187*x + 6561, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 3 x^{14} - 8 x^{13} + 8 x^{12} + 7 x^{11} + 6 x^{10} + 56 x^{9} - 137 x^{8} + 168 x^{7} + 54 x^{6} + 189 x^{5} + 648 x^{4} - 1944 x^{3} + 2187 x^{2} - 2187 x + 6561 \)

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:  \(343361479062744140625=3^{8}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.21$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(1,·)$, $\chi_{165}(67,·)$, $\chi_{165}(133,·)$, $\chi_{165}(76,·)$, $\chi_{165}(34,·)$, $\chi_{165}(142,·)$, $\chi_{165}(131,·)$, $\chi_{165}(23,·)$, $\chi_{165}(89,·)$, $\chi_{165}(32,·)$, $\chi_{165}(98,·)$, $\chi_{165}(164,·)$, $\chi_{165}(43,·)$, $\chi_{165}(109,·)$, $\chi_{165}(56,·)$, $\chi_{165}(122,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{11} - \frac{1}{27} a^{10} + \frac{1}{9} a^{9} - \frac{8}{27} a^{8} + \frac{8}{27} a^{7} + \frac{7}{27} a^{6} + \frac{2}{9} a^{5} + \frac{2}{27} a^{4} - \frac{2}{27} a^{3} + \frac{2}{9} a^{2}$, $\frac{1}{162} a^{12} + \frac{1}{81} a^{11} + \frac{1}{162} a^{9} - \frac{8}{81} a^{8} - \frac{25}{81} a^{7} + \frac{1}{6} a^{6} + \frac{37}{81} a^{5} - \frac{25}{81} a^{4} - \frac{1}{2} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{174474} a^{13} - \frac{203}{87237} a^{12} + \frac{248}{29079} a^{11} - \frac{2303}{174474} a^{10} + \frac{3244}{87237} a^{9} - \frac{22825}{87237} a^{8} - \frac{21391}{58158} a^{7} - \frac{3239}{87237} a^{6} - \frac{21655}{87237} a^{5} - \frac{21391}{58158} a^{4} + \frac{320}{1077} a^{3} + \frac{1226}{3231} a^{2} - \frac{7}{2154} a + \frac{65}{359}$, $\frac{1}{523422} a^{14} - \frac{1}{523422} a^{13} - \frac{232}{87237} a^{12} - \frac{7091}{523422} a^{11} + \frac{4301}{523422} a^{10} - \frac{24022}{261711} a^{9} + \frac{30149}{174474} a^{8} - \frac{167929}{523422} a^{7} + \frac{120500}{261711} a^{6} + \frac{35873}{174474} a^{5} - \frac{12179}{58158} a^{4} + \frac{1229}{9693} a^{3} - \frac{1421}{6462} a^{2} - \frac{815}{2154} a + \frac{159}{359}$, $\frac{1}{1570266} a^{15} - \frac{1}{1570266} a^{14} + \frac{1}{523422} a^{13} - \frac{787}{785133} a^{12} + \frac{25145}{1570266} a^{11} - \frac{62039}{1570266} a^{10} + \frac{19504}{261711} a^{9} - \frac{302371}{1570266} a^{8} - \frac{46577}{1570266} a^{7} + \frac{57835}{261711} a^{6} - \frac{9383}{19386} a^{5} - \frac{4885}{19386} a^{4} - \frac{362}{3231} a^{3} + \frac{255}{718} a^{2} + \frac{223}{718} a - \frac{313}{718}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{1085}{1570266} a^{15} + \frac{1085}{785133} a^{14} - \frac{35}{19386} a^{13} + \frac{1085}{1570266} a^{12} - \frac{8680}{785133} a^{11} + \frac{33635}{1570266} a^{10} + \frac{1085}{58158} a^{9} - \frac{18013}{785133} a^{8} + \frac{33635}{1570266} a^{7} - \frac{1085}{6462} a^{6} + \frac{33635}{87237} a^{5} + \frac{14105}{58158} a^{4} - \frac{8855}{19386} a^{3} - \frac{5425}{2154} a + \frac{1085}{359} \) (order $30$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 16078.6045103 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{33})\), \(\Q(\sqrt{-15}, \sqrt{33})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{-11}, \sqrt{-15})\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), 4.0.136125.2, 4.4.15125.1, 8.0.741200625.1, \(\Q(\zeta_{15})\), 8.0.18530015625.2, 8.0.18530015625.3, 8.8.18530015625.1, 8.0.228765625.1, 8.0.18530015625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$