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Properties

Label 16.0.73336002145...5929.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 53^{14}$
Root discriminant $73.55$
Ramified primes $3, 53$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![908209, 4677324, 10597823, 13227060, 10357688, 5468811, 1988414, 418743, 4102, -14010, 3220, 1929, 269, 48, 4, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 4*x^14 + 48*x^13 + 269*x^12 + 1929*x^11 + 3220*x^10 - 14010*x^9 + 4102*x^8 + 418743*x^7 + 1988414*x^6 + 5468811*x^5 + 10357688*x^4 + 13227060*x^3 + 10597823*x^2 + 4677324*x + 908209)
 
gp: K = bnfinit(x^16 - 6*x^15 + 4*x^14 + 48*x^13 + 269*x^12 + 1929*x^11 + 3220*x^10 - 14010*x^9 + 4102*x^8 + 418743*x^7 + 1988414*x^6 + 5468811*x^5 + 10357688*x^4 + 13227060*x^3 + 10597823*x^2 + 4677324*x + 908209, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 4 x^{14} + 48 x^{13} + 269 x^{12} + 1929 x^{11} + 3220 x^{10} - 14010 x^{9} + 4102 x^{8} + 418743 x^{7} + 1988414 x^{6} + 5468811 x^{5} + 10357688 x^{4} + 13227060 x^{3} + 10597823 x^{2} + 4677324 x + 908209 \)

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:  \(733360021453910541415126615929=3^{12}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.55$
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:  $3, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4630561133} a^{14} - \frac{289619535}{4630561133} a^{13} + \frac{2274314610}{4630561133} a^{12} + \frac{537286907}{4630561133} a^{11} - \frac{1435752124}{4630561133} a^{10} + \frac{1843374441}{4630561133} a^{9} + \frac{62983993}{4630561133} a^{8} + \frac{1793032651}{4630561133} a^{7} + \frac{2297677937}{4630561133} a^{6} - \frac{50394588}{420960103} a^{5} - \frac{984302670}{4630561133} a^{4} - \frac{1224536802}{4630561133} a^{3} + \frac{127768786}{272385949} a^{2} - \frac{364984161}{4630561133} a - \frac{1248051119}{4630561133}$, $\frac{1}{189816891088831189041614239553655244295131} a^{15} + \frac{408228194093918864033840656676}{11165699475813599355389072914920896723243} a^{14} - \frac{92397193398595633787363557847214475513689}{189816891088831189041614239553655244295131} a^{13} - \frac{45480405715441499192071084935190500577106}{189816891088831189041614239553655244295131} a^{12} - \frac{16360626158864300942674635811443620944121}{189816891088831189041614239553655244295131} a^{11} - \frac{57206109069210391439803267823912783598456}{189816891088831189041614239553655244295131} a^{10} + \frac{8985091097818369860208495404538448182619}{189816891088831189041614239553655244295131} a^{9} + \frac{66316963222271036248404092046940524765084}{189816891088831189041614239553655244295131} a^{8} + \frac{90456680149226917804853211611058342734415}{189816891088831189041614239553655244295131} a^{7} + \frac{6957701647198612154053266997155458463}{1015063588710327214126279355901899702113} a^{6} + \frac{81684089749531329609917362209478835212983}{189816891088831189041614239553655244295131} a^{5} - \frac{62598939281060690491960113382152172076388}{189816891088831189041614239553655244295131} a^{4} + \frac{50288445491057170411159288786470142162926}{189816891088831189041614239553655244295131} a^{3} - \frac{68662680742146447052250954030211550478087}{189816891088831189041614239553655244295131} a^{2} + \frac{48793574172863984205483569292375120130595}{189816891088831189041614239553655244295131} a + \frac{3808548248658268751656000300149903700}{18107115433447599832263115477788347257}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( -\frac{308377976206818509205561507}{40992200650605510328247779458407} a^{15} + \frac{2203501264532479941648757319}{40992200650605510328247779458407} a^{14} - \frac{3753780533065573963068672427}{40992200650605510328247779458407} a^{13} - \frac{10676586673794170188850580137}{40992200650605510328247779458407} a^{12} - \frac{69245926155749848637877476821}{40992200650605510328247779458407} a^{11} - \frac{520679530272428521826595164907}{40992200650605510328247779458407} a^{10} - \frac{389510422383154332374700617220}{40992200650605510328247779458407} a^{9} + \frac{4732978572854749266220722296837}{40992200650605510328247779458407} a^{8} - \frac{6843388862106586740706945788662}{40992200650605510328247779458407} a^{7} - \frac{121059072033237020086670954666756}{40992200650605510328247779458407} a^{6} - \frac{472725822138739034962147615722798}{40992200650605510328247779458407} a^{5} - \frac{1151694829181824520256960274987827}{40992200650605510328247779458407} a^{4} - \frac{1908533361256396403885066057093747}{40992200650605510328247779458407} a^{3} - \frac{1991309819383576637232134744337467}{40992200650605510328247779458407} a^{2} - \frac{1184402622413092176754774463467044}{40992200650605510328247779458407} a - \frac{298960548995815984907326522806}{43013851679544082191235865119} \) (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:  \( 57102345.4203 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

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 $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-159}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{53})\), 4.4.1339893.1, 4.0.148877.1, 8.0.1795313251449.1, 8.4.856364420941173.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
53Data not computed