↑

Properties

Label 16.0.23569008217...8656.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 11^{8}$
Root discriminant $18.76$
Ramified primes $2, 11$
Class number $1$
Class group Trivial
Galois group $Q_8 : C_2$ (as 16T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89, -384, 580, -492, 776, -1532, 2316, -2092, 1013, -4, -360, 228, -38, -28, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 - 38*x^12 + 228*x^11 - 360*x^10 - 4*x^9 + 1013*x^8 - 2092*x^7 + 2316*x^6 - 1532*x^5 + 776*x^4 - 492*x^3 + 580*x^2 - 384*x + 89)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 28*x^13 - 38*x^12 + 228*x^11 - 360*x^10 - 4*x^9 + 1013*x^8 - 2092*x^7 + 2316*x^6 - 1532*x^5 + 776*x^4 - 492*x^3 + 580*x^2 - 384*x + 89, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 28 x^{13} - 38 x^{12} + 228 x^{11} - 360 x^{10} - 4 x^{9} + 1013 x^{8} - 2092 x^{7} + 2316 x^{6} - 1532 x^{5} + 776 x^{4} - 492 x^{3} + 580 x^{2} - 384 x + 89 \)

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:  \(235690082176551878656=2^{40}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.76$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{15} a^{10} - \frac{1}{3} a^{9} - \frac{7}{15} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{4}{15} a^{2} - \frac{2}{15}$, $\frac{1}{15} a^{11} + \frac{1}{3} a^{9} - \frac{7}{15} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{2}{5} a^{3} - \frac{1}{3} a^{2} - \frac{2}{15} a + \frac{1}{3}$, $\frac{1}{45} a^{12} + \frac{1}{45} a^{10} - \frac{2}{9} a^{9} - \frac{7}{45} a^{8} - \frac{4}{9} a^{7} + \frac{8}{45} a^{6} + \frac{2}{9} a^{5} - \frac{4}{45} a^{4} + \frac{1}{9} a^{3} - \frac{16}{45} a^{2} + \frac{4}{9} a + \frac{8}{45}$, $\frac{1}{315} a^{13} - \frac{1}{105} a^{12} - \frac{1}{63} a^{11} + \frac{8}{315} a^{10} + \frac{113}{315} a^{9} - \frac{44}{315} a^{8} + \frac{4}{63} a^{7} + \frac{94}{315} a^{6} - \frac{4}{315} a^{5} + \frac{32}{315} a^{4} + \frac{83}{315} a^{3} - \frac{31}{315} a^{2} + \frac{19}{63} a - \frac{47}{105}$, $\frac{1}{6139665} a^{14} - \frac{1}{877095} a^{13} - \frac{260}{58473} a^{12} + \frac{23413}{877095} a^{11} + \frac{153311}{6139665} a^{10} - \frac{756352}{2046555} a^{9} + \frac{7429}{175419} a^{8} + \frac{299719}{877095} a^{7} - \frac{328145}{1227933} a^{6} - \frac{1474747}{6139665} a^{5} + \frac{295138}{6139665} a^{4} - \frac{95554}{1227933} a^{3} - \frac{2200217}{6139665} a^{2} - \frac{1020232}{6139665} a - \frac{24124}{68985}$, $\frac{1}{485033535} a^{15} + \frac{32}{485033535} a^{14} + \frac{23523}{17964205} a^{13} - \frac{1041664}{97006707} a^{12} - \frac{46472}{5449815} a^{11} + \frac{611041}{23096835} a^{10} + \frac{144918173}{485033535} a^{9} + \frac{9686681}{97006707} a^{8} - \frac{189261022}{485033535} a^{7} - \frac{138378853}{485033535} a^{6} + \frac{242337184}{485033535} a^{5} - \frac{124254419}{485033535} a^{4} + \frac{144138577}{485033535} a^{3} + \frac{4833874}{13858101} a^{2} + \frac{142858087}{485033535} a + \frac{97133}{363321}$

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:  \( \frac{39932}{1816605} a^{15} - \frac{5013737}{32335569} a^{14} + \frac{12114134}{32335569} a^{13} - \frac{815100}{3592841} a^{12} - \frac{58727776}{53892615} a^{11} + \frac{631065469}{161677845} a^{10} - \frac{126187037}{32335569} a^{9} - \frac{15390757}{3592841} a^{8} + \frac{318824658}{17964205} a^{7} - \frac{4387474013}{161677845} a^{6} + \frac{3811794929}{161677845} a^{5} - \frac{366009628}{32335569} a^{4} + \frac{1126492858}{161677845} a^{3} - \frac{652598671}{161677845} a^{2} + \frac{143273617}{23096835} a - \frac{3514897}{1816605} \) (order $8$)
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:  \( 16135.3882167 \)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(i, \sqrt{22})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{11})\), \(\Q(\sqrt{-2}, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{11})\), 8.0.959512576.1, 8.0.126877696.1 x2, 8.0.959512576.2 x2, 8.4.15352201216.1 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$