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Properties

Label 16.0.13066428858...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 13^{8}$
Root discriminant $20.88$
Ramified primes $3, 5, 13$
Class number $4$ (GRH)
Class group $[4]$ (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![16, 112, 708, 620, 725, 563, 519, 440, 109, 165, 66, 21, 25, 0, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 7*x^14 + 25*x^12 + 21*x^11 + 66*x^10 + 165*x^9 + 109*x^8 + 440*x^7 + 519*x^6 + 563*x^5 + 725*x^4 + 620*x^3 + 708*x^2 + 112*x + 16)
 
gp: K = bnfinit(x^16 - x^15 + 7*x^14 + 25*x^12 + 21*x^11 + 66*x^10 + 165*x^9 + 109*x^8 + 440*x^7 + 519*x^6 + 563*x^5 + 725*x^4 + 620*x^3 + 708*x^2 + 112*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 7 x^{14} + 25 x^{12} + 21 x^{11} + 66 x^{10} + 165 x^{9} + 109 x^{8} + 440 x^{7} + 519 x^{6} + 563 x^{5} + 725 x^{4} + 620 x^{3} + 708 x^{2} + 112 x + 16 \)

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:  \(1306642885859619140625=3^{8}\cdot 5^{12}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.88$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{5}$, $\frac{1}{48} a^{12} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} + \frac{1}{48} a^{9} - \frac{1}{24} a^{8} - \frac{1}{4} a^{7} + \frac{7}{48} a^{6} + \frac{1}{24} a^{5} + \frac{1}{24} a^{4} - \frac{3}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{12} a + \frac{1}{3}$, $\frac{1}{48} a^{13} - \frac{1}{16} a^{10} + \frac{1}{24} a^{8} - \frac{5}{48} a^{7} - \frac{1}{6} a^{6} - \frac{5}{48} a^{4} + \frac{7}{24} a^{2} - \frac{1}{12} a + \frac{1}{6}$, $\frac{1}{384} a^{14} - \frac{1}{128} a^{13} - \frac{3}{128} a^{11} - \frac{7}{128} a^{10} + \frac{5}{96} a^{9} - \frac{5}{384} a^{8} + \frac{91}{384} a^{7} - \frac{1}{16} a^{6} + \frac{25}{384} a^{5} - \frac{17}{128} a^{4} - \frac{19}{96} a^{3} - \frac{19}{96} a^{2} - \frac{1}{24} a + \frac{3}{8}$, $\frac{1}{22448679936} a^{15} + \frac{1296603}{1870723328} a^{14} + \frac{2267291}{22448679936} a^{13} + \frac{50953069}{7482893312} a^{12} - \frac{186539}{1870723328} a^{11} + \frac{740472641}{22448679936} a^{10} + \frac{1096924615}{22448679936} a^{9} - \frac{106794239}{2806084992} a^{8} - \frac{2075672107}{22448679936} a^{7} + \frac{796670859}{7482893312} a^{6} + \frac{404793273}{1870723328} a^{5} + \frac{1708471375}{22448679936} a^{4} - \frac{51544679}{701521248} a^{3} + \frac{641517297}{1870723328} a^{2} + \frac{80380507}{350760624} a - \frac{262914893}{1403042496}$

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:  \( -\frac{76918641}{7482893312} a^{15} + \frac{73032475}{5612169984} a^{14} - \frac{572481955}{7482893312} a^{13} + \frac{489355835}{22448679936} a^{12} - \frac{1547036155}{5612169984} a^{11} - \frac{3017434027}{22448679936} a^{10} - \frac{15758671205}{22448679936} a^{9} - \frac{705052565}{467680832} a^{8} - \frac{20348427815}{22448679936} a^{7} - \frac{99509029555}{22448679936} a^{6} - \frac{25264221763}{5612169984} a^{5} - \frac{121878098885}{22448679936} a^{4} - \frac{2345356615}{350760624} a^{3} - \frac{32680917265}{5612169984} a^{2} - \frac{829654085}{116920208} a - \frac{175145521}{1403042496} \) (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:  \( 33349.777132 \) (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

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{65})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{13})\), 8.0.1445900625.1, 8.0.36147515625.4 x2, 8.4.4016390625.1 x2, 8.0.213890625.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$