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Properties

Label 16.0.85844307439...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 181^{6}$
Root discriminant $23.49$
Ramified primes $5, 181$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -459, 1089, -1089, -386, 1879, -1002, -482, 681, -80, -42, -12, -1, 20, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 + 20*x^13 - x^12 - 12*x^11 - 42*x^10 - 80*x^9 + 681*x^8 - 482*x^7 - 1002*x^6 + 1879*x^5 - 386*x^4 - 1089*x^3 + 1089*x^2 - 459*x + 81)
 
gp: K = bnfinit(x^16 - 2*x^15 - 5*x^14 + 20*x^13 - x^12 - 12*x^11 - 42*x^10 - 80*x^9 + 681*x^8 - 482*x^7 - 1002*x^6 + 1879*x^5 - 386*x^4 - 1089*x^3 + 1089*x^2 - 459*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 5 x^{14} + 20 x^{13} - x^{12} - 12 x^{11} - 42 x^{10} - 80 x^{9} + 681 x^{8} - 482 x^{7} - 1002 x^{6} + 1879 x^{5} - 386 x^{4} - 1089 x^{3} + 1089 x^{2} - 459 x + 81 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8584430743916259765625=5^{12}\cdot 181^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.49$
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:  $5, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{2}{9} a^{12} - \frac{4}{9} a^{11} - \frac{4}{9} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{9} a^{7} + \frac{4}{9} a^{5} - \frac{2}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{269726363277673430367} a^{15} + \frac{13915915847538234898}{269726363277673430367} a^{14} - \frac{2495614042297176479}{269726363277673430367} a^{13} + \frac{110255905028184057413}{269726363277673430367} a^{12} + \frac{79477842341864559332}{269726363277673430367} a^{11} + \frac{6724794822629731130}{29969595919741492263} a^{10} - \frac{22787619487736767871}{89908787759224476789} a^{9} + \frac{46593091858177824508}{269726363277673430367} a^{8} + \frac{35029254880222499743}{89908787759224476789} a^{7} + \frac{55421420389025110546}{269726363277673430367} a^{6} - \frac{14356721460944456144}{89908787759224476789} a^{5} + \frac{43988026241414130379}{269726363277673430367} a^{4} - \frac{78593514675971483153}{269726363277673430367} a^{3} - \frac{18731237467201940992}{89908787759224476789} a^{2} + \frac{5868787682149567028}{29969595919741492263} a - \frac{1740566838406173403}{9989865306580497421}$

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:  \( \frac{3433281994457538368}{269726363277673430367} a^{15} - \frac{5503829630558682961}{269726363277673430367} a^{14} - \frac{18731391854287311340}{269726363277673430367} a^{13} + \frac{60419009814054795973}{269726363277673430367} a^{12} + \frac{17663468919411794296}{269726363277673430367} a^{11} - \frac{8651463567167375684}{89908787759224476789} a^{10} - \frac{50706496390001297032}{89908787759224476789} a^{9} - \frac{324273628477408971217}{269726363277673430367} a^{8} + \frac{726863904665304290032}{89908787759224476789} a^{7} - \frac{852643893918418065052}{269726363277673430367} a^{6} - \frac{1153218169136850692249}{89908787759224476789} a^{5} + \frac{4957067474367841681604}{269726363277673430367} a^{4} + \frac{502252115630327093093}{269726363277673430367} a^{3} - \frac{118387237231008419325}{9989865306580497421} a^{2} + \frac{91294933617019402024}{9989865306580497421} a - \frac{17019425584767442571}{9989865306580497421} \) (order $10$)
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:  \( 43749.6863738 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

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 $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.4525.1, 4.0.22625.1, 8.0.3706088125.1, 8.8.92652203125.1, 8.0.511890625.1

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.6.1$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}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$