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Properties

Label 16.0.21767823360...0000.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{12}$
Root discriminant $21.56$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $Q_8 : C_2$ (as 16T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![441, 1134, 117, -1656, 648, 1614, -2094, 342, 1123, -1206, 604, -150, 18, -12, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 13*x^14 - 12*x^13 + 18*x^12 - 150*x^11 + 604*x^10 - 1206*x^9 + 1123*x^8 + 342*x^7 - 2094*x^6 + 1614*x^5 + 648*x^4 - 1656*x^3 + 117*x^2 + 1134*x + 441)
 
gp: K = bnfinit(x^16 - 6*x^15 + 13*x^14 - 12*x^13 + 18*x^12 - 150*x^11 + 604*x^10 - 1206*x^9 + 1123*x^8 + 342*x^7 - 2094*x^6 + 1614*x^5 + 648*x^4 - 1656*x^3 + 117*x^2 + 1134*x + 441, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 13 x^{14} - 12 x^{13} + 18 x^{12} - 150 x^{11} + 604 x^{10} - 1206 x^{9} + 1123 x^{8} + 342 x^{7} - 2094 x^{6} + 1614 x^{5} + 648 x^{4} - 1656 x^{3} + 117 x^{2} + 1134 x + 441 \)

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:  \(2176782336000000000000=2^{24}\cdot 3^{12}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.56$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9} - \frac{9}{19} a^{8} - \frac{3}{19} a^{7} + \frac{5}{19} a^{6} - \frac{8}{19} a^{5} + \frac{8}{19} a^{4} - \frac{9}{19} a^{3} - \frac{2}{19} a^{2} - \frac{5}{19} a + \frac{3}{19}$, $\frac{1}{19} a^{10} - \frac{8}{19} a^{8} - \frac{3}{19} a^{7} - \frac{1}{19} a^{6} - \frac{7}{19} a^{5} + \frac{6}{19} a^{4} - \frac{7}{19} a^{3} - \frac{4}{19} a^{2} - \frac{4}{19} a + \frac{8}{19}$, $\frac{1}{19} a^{11} + \frac{1}{19} a^{8} - \frac{6}{19} a^{7} - \frac{5}{19} a^{6} - \frac{1}{19} a^{5} - \frac{1}{19} a^{2} + \frac{6}{19} a + \frac{5}{19}$, $\frac{1}{798} a^{12} + \frac{1}{133} a^{11} - \frac{1}{399} a^{10} - \frac{1}{133} a^{9} + \frac{115}{266} a^{8} - \frac{34}{133} a^{7} - \frac{311}{798} a^{6} - \frac{40}{133} a^{5} - \frac{11}{798} a^{4} + \frac{1}{7} a^{3} - \frac{9}{133} a^{2} - \frac{5}{133} a - \frac{13}{38}$, $\frac{1}{798} a^{13} + \frac{2}{399} a^{11} + \frac{1}{133} a^{10} + \frac{1}{266} a^{9} + \frac{62}{133} a^{8} + \frac{199}{798} a^{7} + \frac{54}{133} a^{6} - \frac{377}{798} a^{5} + \frac{58}{133} a^{4} + \frac{45}{133} a^{3} + \frac{5}{19} a^{2} - \frac{115}{266} a - \frac{2}{19}$, $\frac{1}{24738} a^{14} - \frac{5}{8246} a^{13} - \frac{13}{24738} a^{12} + \frac{79}{4123} a^{11} + \frac{325}{24738} a^{10} + \frac{73}{8246} a^{9} + \frac{158}{399} a^{8} + \frac{163}{434} a^{7} + \frac{2524}{12369} a^{6} - \frac{3821}{8246} a^{5} - \frac{10559}{24738} a^{4} - \frac{1089}{4123} a^{3} - \frac{635}{8246} a^{2} + \frac{313}{8246} a - \frac{291}{1178}$, $\frac{1}{37131738} a^{15} - \frac{283}{18565869} a^{14} - \frac{4241}{37131738} a^{13} - \frac{19147}{37131738} a^{12} + \frac{266409}{12377246} a^{11} + \frac{27473}{18565869} a^{10} + \frac{120752}{18565869} a^{9} - \frac{3877427}{37131738} a^{8} - \frac{6658369}{18565869} a^{7} - \frac{13498141}{37131738} a^{6} - \frac{4506863}{12377246} a^{5} - \frac{1729523}{37131738} a^{4} + \frac{2713731}{12377246} a^{3} + \frac{81324}{6188623} a^{2} - \frac{740653}{12377246} a + \frac{279505}{1768178}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{3926}{235011} a^{15} + \frac{3880}{33573} a^{14} - \frac{74660}{235011} a^{13} + \frac{109285}{235011} a^{12} - \frac{7590}{11191} a^{11} + \frac{241061}{78337} a^{10} - \frac{3005360}{235011} a^{9} + \frac{7307500}{235011} a^{8} - \frac{1499270}{33573} a^{7} + \frac{7157845}{235011} a^{6} + \frac{934644}{78337} a^{5} - \frac{3033785}{78337} a^{4} + \frac{1664640}{78337} a^{3} + \frac{883385}{78337} a^{2} - \frac{143160}{11191} a - \frac{99151}{11191} \) (order $6$)
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:  \( 26643.8469845 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_2$ (as 16T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $Q_8 : C_2$
Character table for $Q_8 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), 8.0.207360000.2, 8.0.729000000.1 x2, 8.0.46656000000.1 x2, 8.4.46656000000.2 x2

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

Sibling fields

Degree 8 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 R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$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}$