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Properties

Label 16.0.25991671039...5696.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{8}\cdot 7^{8}$
Root discriminant $21.80$
Ramified primes $2, 3, 7$
Class number $4$
Class group $[2, 2]$
Galois group $Q_8 : C_2^2$ (as 16T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![58, -816, 4880, -5196, -1128, 3708, -562, -960, 711, 228, -214, 24, 68, -12, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 - 12*x^13 + 68*x^12 + 24*x^11 - 214*x^10 + 228*x^9 + 711*x^8 - 960*x^7 - 562*x^6 + 3708*x^5 - 1128*x^4 - 5196*x^3 + 4880*x^2 - 816*x + 58)
 
gp: K = bnfinit(x^16 - 12*x^14 - 12*x^13 + 68*x^12 + 24*x^11 - 214*x^10 + 228*x^9 + 711*x^8 - 960*x^7 - 562*x^6 + 3708*x^5 - 1128*x^4 - 5196*x^3 + 4880*x^2 - 816*x + 58, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{14} - 12 x^{13} + 68 x^{12} + 24 x^{11} - 214 x^{10} + 228 x^{9} + 711 x^{8} - 960 x^{7} - 562 x^{6} + 3708 x^{5} - 1128 x^{4} - 5196 x^{3} + 4880 x^{2} - 816 x + 58 \)

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:  \(2599167103947239325696=2^{36}\cdot 3^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.80$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{35} a^{12} + \frac{13}{35} a^{11} - \frac{8}{35} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{13}{35} a^{7} + \frac{8}{35} a^{6} + \frac{12}{35} a^{5} - \frac{8}{35} a^{2} - \frac{13}{35} a - \frac{12}{35}$, $\frac{1}{35} a^{13} - \frac{2}{35} a^{11} - \frac{16}{35} a^{10} - \frac{1}{7} a^{9} - \frac{12}{35} a^{8} + \frac{2}{5} a^{7} + \frac{13}{35} a^{6} - \frac{16}{35} a^{5} - \frac{8}{35} a^{3} - \frac{2}{5} a^{2} + \frac{17}{35} a + \frac{16}{35}$, $\frac{1}{1085} a^{14} + \frac{6}{1085} a^{13} + \frac{9}{1085} a^{12} + \frac{9}{217} a^{11} - \frac{2}{155} a^{10} - \frac{487}{1085} a^{9} - \frac{508}{1085} a^{8} - \frac{64}{217} a^{7} - \frac{26}{217} a^{6} + \frac{456}{1085} a^{5} + \frac{342}{1085} a^{4} + \frac{253}{1085} a^{3} - \frac{80}{217} a^{2} - \frac{40}{217} a + \frac{349}{1085}$, $\frac{1}{32983342810385158370165} a^{15} - \frac{929457650444588081}{6596668562077031674033} a^{14} + \frac{267588587362877129224}{32983342810385158370165} a^{13} + \frac{441949323035177175096}{32983342810385158370165} a^{12} - \frac{11122985901282360897156}{32983342810385158370165} a^{11} - \frac{8066061083046096906114}{32983342810385158370165} a^{10} + \frac{27647463327489131762}{151996971476429301245} a^{9} - \frac{1371097776399778095159}{32983342810385158370165} a^{8} + \frac{9768658508369136601894}{32983342810385158370165} a^{7} + \frac{8204350744262710287834}{32983342810385158370165} a^{6} - \frac{98653012265214260226}{6596668562077031674033} a^{5} + \frac{14060146246719270321796}{32983342810385158370165} a^{4} + \frac{1981763347891196617107}{4711906115769308338595} a^{3} + \frac{11219084531871345527726}{32983342810385158370165} a^{2} - \frac{15894853306492185941784}{32983342810385158370165} a + \frac{16138137615025685458712}{32983342810385158370165}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( -\frac{15346061088}{3723133478915} a^{15} - \frac{2707912112}{3723133478915} a^{14} + \frac{36381931686}{744626695783} a^{13} + \frac{215341679189}{3723133478915} a^{12} - \frac{197209093332}{744626695783} a^{11} - \frac{510525483579}{3723133478915} a^{10} + \frac{3100231404462}{3723133478915} a^{9} - \frac{607913635498}{744626695783} a^{8} - \frac{11178808724076}{3723133478915} a^{7} + \frac{2523485295491}{744626695783} a^{6} + \frac{9722570264754}{3723133478915} a^{5} - \frac{54659761856027}{3723133478915} a^{4} + \frac{1647865992930}{744626695783} a^{3} + \frac{77015422983596}{3723133478915} a^{2} - \frac{12607902987732}{744626695783} a + \frac{1149830096473}{744626695783} \) (order $4$)
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:  \( 21961.6900705 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2$ (as 16T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(i, \sqrt{42})\), \(\Q(\sqrt{-6}, \sqrt{7})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(i, \sqrt{7})\), 8.0.12745506816.7, 8.0.1040449536.1 x2, 8.0.354041856.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.18.55$x^{8} + 2 x^{6} + 4 x^{3} + 10$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.18.55$x^{8} + 2 x^{6} + 4 x^{3} + 10$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
$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}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$