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Properties

Label 16.8.81925410828...5625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{14}\cdot 41^{10}$
Root discriminant $41.65$
Ramified primes $5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 270, 738, 450, -1432, 2230, 896, -4760, 1680, 1450, -1559, 715, -7, -45, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^14 - 45*x^13 - 7*x^12 + 715*x^11 - 1559*x^10 + 1450*x^9 + 1680*x^8 - 4760*x^7 + 896*x^6 + 2230*x^5 - 1432*x^4 + 450*x^3 + 738*x^2 + 270*x + 81)
 
gp: K = bnfinit(x^16 - 7*x^14 - 45*x^13 - 7*x^12 + 715*x^11 - 1559*x^10 + 1450*x^9 + 1680*x^8 - 4760*x^7 + 896*x^6 + 2230*x^5 - 1432*x^4 + 450*x^3 + 738*x^2 + 270*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{14} - 45 x^{13} - 7 x^{12} + 715 x^{11} - 1559 x^{10} + 1450 x^{9} + 1680 x^{8} - 4760 x^{7} + 896 x^{6} + 2230 x^{5} - 1432 x^{4} + 450 x^{3} + 738 x^{2} + 270 x + 81 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(81925410828566900634765625=5^{14}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.65$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{3} a^{6} - \frac{1}{9} a^{5} - \frac{4}{9} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{14} - \frac{1}{27} a^{13} + \frac{1}{9} a^{11} - \frac{1}{27} a^{10} - \frac{4}{27} a^{9} - \frac{4}{27} a^{8} - \frac{10}{27} a^{7} + \frac{10}{27} a^{6} - \frac{2}{9} a^{5} + \frac{8}{27} a^{4} - \frac{10}{27} a^{3} + \frac{4}{9} a^{2}$, $\frac{1}{425467719814347083416023} a^{15} - \frac{1422950191688773715038}{141822573271449027805341} a^{14} - \frac{23005321982818941982471}{425467719814347083416023} a^{13} + \frac{2116460153658778444459}{141822573271449027805341} a^{12} + \frac{29591417179517604412448}{425467719814347083416023} a^{11} - \frac{4111141770277321366469}{425467719814347083416023} a^{10} - \frac{5632878929870250078674}{425467719814347083416023} a^{9} + \frac{25503544697090378879689}{425467719814347083416023} a^{8} + \frac{68195156956026622222501}{141822573271449027805341} a^{7} - \frac{63596850549169062469193}{425467719814347083416023} a^{6} + \frac{18767727532549295572322}{425467719814347083416023} a^{5} - \frac{139532216814367889227595}{425467719814347083416023} a^{4} + \frac{14265589312409053860752}{425467719814347083416023} a^{3} - \frac{13988867175975917782574}{141822573271449027805341} a^{2} - \frac{87021745160654676890}{1750895966314185528461} a - \frac{346465588316416374626}{15758063696827669756149}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 23009235.0124 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{205}) \), \(\Q(\sqrt{41}) \), 4.4.5125.1 x2, 4.4.210125.1 x2, \(\Q(\sqrt{5}, \sqrt{41})\), 8.4.9051265703125.2, 8.4.5384453125.2, 8.8.44152515625.1

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 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$