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Properties

Label 16.8.64479545841...7129.1
Degree $16$
Signature $[8, 4]$
Discriminant $61^{14}\cdot 97^{14}$
Root discriminant $1997.97$
Ramified primes $61, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15827972949927, 16445814982743, -13035184353877, -16295183935748, -2180629500814, 878429017229, -130866147686, -19389915190, 10642508325, 18163842, 2357862, -4647376, -1000472, 98, -23, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 23*x^14 + 98*x^13 - 1000472*x^12 - 4647376*x^11 + 2357862*x^10 + 18163842*x^9 + 10642508325*x^8 - 19389915190*x^7 - 130866147686*x^6 + 878429017229*x^5 - 2180629500814*x^4 - 16295183935748*x^3 - 13035184353877*x^2 + 16445814982743*x + 15827972949927)
 
gp: K = bnfinit(x^16 - 4*x^15 - 23*x^14 + 98*x^13 - 1000472*x^12 - 4647376*x^11 + 2357862*x^10 + 18163842*x^9 + 10642508325*x^8 - 19389915190*x^7 - 130866147686*x^6 + 878429017229*x^5 - 2180629500814*x^4 - 16295183935748*x^3 - 13035184353877*x^2 + 16445814982743*x + 15827972949927, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 23 x^{14} + 98 x^{13} - 1000472 x^{12} - 4647376 x^{11} + 2357862 x^{10} + 18163842 x^{9} + 10642508325 x^{8} - 19389915190 x^{7} - 130866147686 x^{6} + 878429017229 x^{5} - 2180629500814 x^{4} - 16295183935748 x^{3} - 13035184353877 x^{2} + 16445814982743 x + 15827972949927 \)

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:  \(64479545841814254728122854912988442937092177855107129=61^{14}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1997.97$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{183} a^{8} - \frac{2}{183} a^{7} + \frac{17}{183} a^{6} + \frac{22}{183} a^{5} - \frac{15}{61} a^{4} + \frac{3}{61} a^{3} - \frac{85}{183} a^{2} + \frac{18}{61} a - \frac{12}{61}$, $\frac{1}{183} a^{9} + \frac{13}{183} a^{7} - \frac{5}{183} a^{6} - \frac{1}{183} a^{5} - \frac{20}{183} a^{4} - \frac{2}{61} a^{3} + \frac{2}{61} a^{2} - \frac{50}{183} a - \frac{24}{61}$, $\frac{1}{183} a^{10} + \frac{7}{61} a^{7} + \frac{22}{183} a^{6} + \frac{20}{61} a^{5} - \frac{31}{183} a^{4} + \frac{11}{183} a^{3} + \frac{6}{61} a^{2} + \frac{80}{183} a - \frac{27}{61}$, $\frac{1}{1647} a^{11} + \frac{1}{549} a^{10} + \frac{1}{1647} a^{9} + \frac{4}{1647} a^{8} + \frac{44}{549} a^{7} - \frac{56}{549} a^{6} + \frac{140}{1647} a^{5} - \frac{206}{549} a^{4} + \frac{208}{549} a^{3} - \frac{428}{1647} a^{2} + \frac{472}{1647} a - \frac{206}{549}$, $\frac{1}{1647} a^{12} + \frac{1}{1647} a^{10} + \frac{1}{1647} a^{9} + \frac{1}{549} a^{8} - \frac{47}{549} a^{7} - \frac{49}{1647} a^{6} + \frac{74}{549} a^{5} - \frac{74}{549} a^{4} + \frac{589}{1647} a^{3} - \frac{215}{1647} a^{2} - \frac{55}{183} a + \frac{44}{183}$, $\frac{1}{1647} a^{13} - \frac{2}{1647} a^{10} + \frac{2}{1647} a^{9} - \frac{1}{1647} a^{8} + \frac{80}{1647} a^{7} + \frac{31}{549} a^{6} + \frac{10}{27} a^{5} + \frac{217}{1647} a^{4} + \frac{457}{1647} a^{3} - \frac{778}{1647} a^{2} + \frac{563}{1647} a + \frac{125}{549}$, $\frac{1}{11134682771850063} a^{14} - \frac{2951864156893}{11134682771850063} a^{13} + \frac{81171231637}{412395658216669} a^{12} - \frac{76604471168}{3711560923950021} a^{11} - \frac{23413051809173}{11134682771850063} a^{10} - \frac{16494438041356}{11134682771850063} a^{9} - \frac{9869117880658}{11134682771850063} a^{8} - \frac{147724845233426}{11134682771850063} a^{7} - \frac{1443302526718637}{11134682771850063} a^{6} + \frac{4902991115685334}{11134682771850063} a^{5} + \frac{5352058840408}{20281753682787} a^{4} - \frac{1028702866490639}{11134682771850063} a^{3} - \frac{3921381228828973}{11134682771850063} a^{2} - \frac{1564656093974995}{3711560923950021} a + \frac{100590744450973}{412395658216669}$, $\frac{1}{5373572679191851822176534360931351220057426788703486869846957395770300648222697} a^{15} - \frac{7384699550706936079094630182488688055568710850220077277860266}{199021210340438956376908680034494489631756547729758772957294718361862986971211} a^{14} - \frac{1464512172535864089034712593930956628836791926971789453847770674223771693386}{5373572679191851822176534360931351220057426788703486869846957395770300648222697} a^{13} + \frac{501743843261195679809786383920889077880068513846668431541748099864404805692}{1791190893063950607392178120310450406685808929567828956615652465256766882740899} a^{12} + \frac{576946662893820603584158213778574875325892553755140506562304060759197175534}{5373572679191851822176534360931351220057426788703486869846957395770300648222697} a^{11} - \frac{4617450475475626330435149500977331241803720430931116627207238498973544924685}{5373572679191851822176534360931351220057426788703486869846957395770300648222697} a^{10} - \frac{602859892999159189897008580057278764837877347596658363384226649213440089437}{5373572679191851822176534360931351220057426788703486869846957395770300648222697} a^{9} + \frac{12720018652996597686500001107545718187390800473307238834153490463291758776049}{5373572679191851822176534360931351220057426788703486869846957395770300648222697} a^{8} - \frac{32018997126555296683580030159587576753843365858115855551608524357207338119423}{1791190893063950607392178120310450406685808929567828956615652465256766882740899} a^{7} + \frac{33078364321307410902096370756016546211029611830541365915815983881860333582981}{5373572679191851822176534360931351220057426788703486869846957395770300648222697} a^{6} - \frac{52683923696400396679259342793100704046129215739249321013896061909846078153793}{199021210340438956376908680034494489631756547729758772957294718361862986971211} a^{5} - \frac{200777028183297045071128990944484759745624103482700630673968193106501800712576}{1791190893063950607392178120310450406685808929567828956615652465256766882740899} a^{4} - \frac{656211647316963618745575744687658280767954226525202383064902623552317159280006}{5373572679191851822176534360931351220057426788703486869846957395770300648222697} a^{3} + \frac{15749611234495656256062297074544353575788574044833053093694712042574812502399}{199021210340438956376908680034494489631756547729758772957294718361862986971211} a^{2} + \frac{2243237773148005026390166110512365220765317579261639343249138901510148405674544}{5373572679191851822176534360931351220057426788703486869846957395770300648222697} a + \frac{185867475955417909749272701459038636860628243179573004161865495895861392907678}{1791190893063950607392178120310450406685808929567828956615652465256766882740899}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 119686659626000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

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 $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{97}) \), \(\Q(\sqrt{5917}) \), \(\Q(\sqrt{61}, \sqrt{97})\), 8.8.42915029526174817225369.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$61$61.8.7.1$x^{8} - 61$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.1$x^{8} - 61$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$97$97.8.7.4$x^{8} - 1515625$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.4$x^{8} - 1515625$$8$$1$$7$$C_8$$[\ ]_{8}$