Properties

Label 16.0.34297420960...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $22.18$
Ramified primes $2, 5, 11$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
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, 0, 3645, 0, 1296, 0, 315, 0, 31, 0, 35, 0, 16, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 5*x^14 + 16*x^12 + 35*x^10 + 31*x^8 + 315*x^6 + 1296*x^4 + 3645*x^2 + 6561)
 
gp: K = bnfinit(x^16 + 5*x^14 + 16*x^12 + 35*x^10 + 31*x^8 + 315*x^6 + 1296*x^4 + 3645*x^2 + 6561, 1)
 

Normalized defining polynomial

\( x^{16} + 5 x^{14} + 16 x^{12} + 35 x^{10} + 31 x^{8} + 315 x^{6} + 1296 x^{4} + 3645 x^{2} + 6561 \)

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:  \(3429742096000000000000=2^{16}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.18$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(220=2^{2}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(67,·)$, $\chi_{220}(133,·)$, $\chi_{220}(199,·)$, $\chi_{220}(23,·)$, $\chi_{220}(131,·)$, $\chi_{220}(21,·)$, $\chi_{220}(87,·)$, $\chi_{220}(153,·)$, $\chi_{220}(219,·)$, $\chi_{220}(197,·)$, $\chi_{220}(43,·)$, $\chi_{220}(109,·)$, $\chi_{220}(111,·)$, $\chi_{220}(177,·)$, $\chi_{220}(89,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{279} a^{10} - \frac{1}{9} a^{8} + \frac{4}{9} a^{6} + \frac{2}{9} a^{4} + \frac{1}{9} a^{2} + \frac{4}{31}$, $\frac{1}{837} a^{11} - \frac{1}{27} a^{9} + \frac{13}{27} a^{7} - \frac{7}{27} a^{5} + \frac{10}{27} a^{3} + \frac{4}{93} a$, $\frac{1}{2511} a^{12} - \frac{4}{2511} a^{10} - \frac{14}{81} a^{8} + \frac{20}{81} a^{6} - \frac{17}{81} a^{4} + \frac{4}{279} a^{2} + \frac{12}{31}$, $\frac{1}{7533} a^{13} - \frac{4}{7533} a^{11} - \frac{14}{243} a^{9} - \frac{61}{243} a^{7} + \frac{64}{243} a^{5} - \frac{275}{837} a^{3} + \frac{4}{31} a$, $\frac{1}{22599} a^{14} - \frac{4}{22599} a^{12} - \frac{29}{22599} a^{10} + \frac{20}{729} a^{8} + \frac{226}{729} a^{6} + \frac{4}{2511} a^{4} - \frac{19}{279} a^{2} - \frac{11}{31}$, $\frac{1}{67797} a^{15} - \frac{4}{67797} a^{13} - \frac{29}{67797} a^{11} + \frac{20}{2187} a^{9} + \frac{955}{2187} a^{7} + \frac{2515}{7533} a^{5} - \frac{298}{837} a^{3} + \frac{20}{93} a$

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

Class group and class number

$C_{4}$, 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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{2}{837} a^{13} + \frac{599}{837} a^{3} \) (order $20$)
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:  \( 40470.7919711 \) (assuming GRH)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-55}) \), \(\Q(i, \sqrt{11})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{55})\), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\sqrt{-5}, \sqrt{11})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{-5}, \sqrt{-11})\), 4.0.242000.2, 4.4.15125.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 8.0.2342560000.1, 8.0.58564000000.2, \(\Q(\zeta_{20})\), 8.0.58564000000.3, 8.8.58564000000.1, 8.0.58564000000.1, 8.0.228765625.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$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}$