Properties

Label 16.0.92340965236...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $15.32$
Ramified primes $3, 5, 7$
Class number $1$
Class group Trivial
Galois group $C_4\times C_2^2$ (as 16T2)

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magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -128, 128, -160, 80, 8, 24, 10, -21, 5, 6, 1, 5, -5, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 2*x^14 - 5*x^13 + 5*x^12 + x^11 + 6*x^10 + 5*x^9 - 21*x^8 + 10*x^7 + 24*x^6 + 8*x^5 + 80*x^4 - 160*x^3 + 128*x^2 - 128*x + 256)
 
gp: K = bnfinit(x^16 - x^15 + 2*x^14 - 5*x^13 + 5*x^12 + x^11 + 6*x^10 + 5*x^9 - 21*x^8 + 10*x^7 + 24*x^6 + 8*x^5 + 80*x^4 - 160*x^3 + 128*x^2 - 128*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 2 x^{14} - 5 x^{13} + 5 x^{12} + x^{11} + 6 x^{10} + 5 x^{9} - 21 x^{8} + 10 x^{7} + 24 x^{6} + 8 x^{5} + 80 x^{4} - 160 x^{3} + 128 x^{2} - 128 x + 256 \)

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:  \(9234096523681640625=3^{8}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.32$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(105=3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(71,·)$, $\chi_{105}(8,·)$, $\chi_{105}(76,·)$, $\chi_{105}(13,·)$, $\chi_{105}(83,·)$, $\chi_{105}(22,·)$, $\chi_{105}(92,·)$, $\chi_{105}(29,·)$, $\chi_{105}(97,·)$, $\chi_{105}(34,·)$, $\chi_{105}(104,·)$, $\chi_{105}(41,·)$, $\chi_{105}(43,·)$, $\chi_{105}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{3}{8} a^{7} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} - \frac{3}{16} a^{8} + \frac{5}{16} a^{7} + \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{3}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{2848} a^{13} - \frac{73}{2848} a^{12} - \frac{51}{1424} a^{11} + \frac{27}{2848} a^{10} + \frac{13}{2848} a^{9} + \frac{777}{2848} a^{8} + \frac{151}{1424} a^{7} - \frac{1083}{2848} a^{6} - \frac{221}{2848} a^{5} + \frac{561}{1424} a^{4} - \frac{37}{356} a^{3} - \frac{27}{356} a^{2} - \frac{1}{178} a - \frac{39}{89}$, $\frac{1}{5696} a^{14} - \frac{1}{5696} a^{13} - \frac{9}{2848} a^{12} + \frac{159}{5696} a^{11} - \frac{179}{5696} a^{10} + \frac{645}{5696} a^{9} - \frac{535}{2848} a^{8} + \frac{1793}{5696} a^{7} + \frac{835}{5696} a^{6} + \frac{259}{2848} a^{5} + \frac{275}{1424} a^{4} + \frac{123}{356} a^{3} - \frac{83}{356} a^{2} + \frac{7}{89} a + \frac{20}{89}$, $\frac{1}{11392} a^{15} - \frac{1}{11392} a^{14} - \frac{1}{5696} a^{13} - \frac{297}{11392} a^{12} + \frac{325}{11392} a^{11} - \frac{347}{11392} a^{10} + \frac{637}{5696} a^{9} - \frac{2151}{11392} a^{8} + \frac{3531}{11392} a^{7} + \frac{851}{5696} a^{6} + \frac{281}{2848} a^{5} + \frac{265}{1424} a^{4} + \frac{61}{178} a^{3} - \frac{47}{178} a^{2} + \frac{8}{89} a + \frac{22}{89}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{1}{356} a^{15} - \frac{3}{356} a^{14} + \frac{3}{178} a^{12} + \frac{7}{356} a^{11} + \frac{11}{356} a^{10} - \frac{4}{89} a^{9} - \frac{11}{356} a^{8} + \frac{17}{178} a^{7} + \frac{8}{89} a^{6} + \frac{22}{89} a^{5} - \frac{20}{89} a^{4} - \frac{8}{89} a^{3} + \frac{271}{356} a^{2} + \frac{32}{89} a + \frac{64}{89} \) (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:  \( 7204.63964805 \)
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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), 4.4.6125.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), 4.0.55125.1, 8.0.121550625.1, 8.0.37515625.1, 8.0.3038765625.2, 8.8.3038765625.1, 8.0.3038765625.3, 8.0.3038765625.1, \(\Q(\zeta_{15})\)

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 R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$