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Properties

Label 16.0.17878103347...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 17^{8}$
Root discriminant $15.97$
Ramified primes $3, 5, 17$
Class number $3$
Class group $[3]$
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![81, 0, -135, 0, 261, 0, -30, 0, 82, 0, 10, 0, 29, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 5*x^14 + 29*x^12 + 10*x^10 + 82*x^8 - 30*x^6 + 261*x^4 - 135*x^2 + 81)
 
gp: K = bnfinit(x^16 + 5*x^14 + 29*x^12 + 10*x^10 + 82*x^8 - 30*x^6 + 261*x^4 - 135*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{16} + 5 x^{14} + 29 x^{12} + 10 x^{10} + 82 x^{8} - 30 x^{6} + 261 x^{4} - 135 x^{2} + 81 \)

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:  \(17878103347812890625=3^{8}\cdot 5^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.97$
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:  $3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{6} - \frac{1}{8} a^{3} + \frac{1}{8}$, $\frac{1}{24} a^{10} + \frac{1}{12} a^{8} - \frac{1}{8} a^{7} + \frac{1}{12} a^{6} - \frac{5}{24} a^{4} - \frac{1}{12} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{1}{8} a^{8} + \frac{1}{12} a^{7} - \frac{1}{8} a^{6} - \frac{5}{24} a^{5} + \frac{1}{24} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{288} a^{12} - \frac{1}{72} a^{10} - \frac{1}{18} a^{8} - \frac{1}{8} a^{7} + \frac{5}{144} a^{6} - \frac{11}{72} a^{4} - \frac{1}{6} a^{2} - \frac{3}{8} a - \frac{3}{32}$, $\frac{1}{864} a^{13} - \frac{1}{54} a^{11} - \frac{1}{216} a^{9} + \frac{29}{432} a^{7} - \frac{1}{8} a^{6} + \frac{1}{54} a^{5} - \frac{5}{72} a^{3} - \frac{19}{96} a + \frac{1}{8}$, $\frac{1}{32832} a^{14} - \frac{1}{1728} a^{13} + \frac{47}{32832} a^{12} + \frac{1}{108} a^{11} + \frac{29}{2052} a^{10} - \frac{13}{216} a^{9} + \frac{1541}{16416} a^{8} - \frac{29}{864} a^{7} + \frac{1151}{16416} a^{6} - \frac{1}{108} a^{5} - \frac{17}{684} a^{4} + \frac{7}{72} a^{3} + \frac{1693}{3648} a^{2} - \frac{77}{192} a + \frac{441}{1216}$, $\frac{1}{98496} a^{15} - \frac{5}{49248} a^{13} - \frac{1}{576} a^{12} + \frac{1}{12312} a^{11} - \frac{1}{72} a^{10} + \frac{1313}{49248} a^{9} - \frac{1}{72} a^{8} + \frac{1117}{24624} a^{7} - \frac{17}{288} a^{6} + \frac{299}{1368} a^{5} + \frac{13}{72} a^{4} + \frac{2149}{10944} a^{3} - \frac{3}{8} a^{2} + \frac{401}{1824} a - \frac{5}{64}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{55}{8208} a^{14} - \frac{553}{16416} a^{12} - \frac{395}{2052} a^{10} - \frac{281}{4104} a^{8} - \frac{4345}{8208} a^{6} - \frac{79}{684} a^{4} - \frac{1535}{912} a^{2} + \frac{529}{608} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1193.88585358 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-3}, \sqrt{85})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-15}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{-51})\), \(\Q(\sqrt{-15}, \sqrt{-51})\), \(\Q(\sqrt{5}, \sqrt{17})\), 8.0.4228250625.1, 8.0.14630625.1 x2, 8.0.169130025.1 x2, 8.4.469805625.1 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$