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Properties

Label 16.0.78664754221...4177.5
Degree $16$
Signature $[0, 8]$
Discriminant $43^{8}\cdot 97^{13}$
Root discriminant $269.77$
Ramified primes $43, 97$
Class number $64$ (GRH)
Class group $[2, 4, 8]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36752433199, -18442347063, 5795387679, -2092377208, 345816880, -13870987, -18720912, 14774315, -3371036, 39761, 136101, -35066, 1534, 850, -101, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 101*x^14 + 850*x^13 + 1534*x^12 - 35066*x^11 + 136101*x^10 + 39761*x^9 - 3371036*x^8 + 14774315*x^7 - 18720912*x^6 - 13870987*x^5 + 345816880*x^4 - 2092377208*x^3 + 5795387679*x^2 - 18442347063*x + 36752433199)
 
gp: K = bnfinit(x^16 - 6*x^15 - 101*x^14 + 850*x^13 + 1534*x^12 - 35066*x^11 + 136101*x^10 + 39761*x^9 - 3371036*x^8 + 14774315*x^7 - 18720912*x^6 - 13870987*x^5 + 345816880*x^4 - 2092377208*x^3 + 5795387679*x^2 - 18442347063*x + 36752433199, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 101 x^{14} + 850 x^{13} + 1534 x^{12} - 35066 x^{11} + 136101 x^{10} + 39761 x^{9} - 3371036 x^{8} + 14774315 x^{7} - 18720912 x^{6} - 13870987 x^{5} + 345816880 x^{4} - 2092377208 x^{3} + 5795387679 x^{2} - 18442347063 x + 36752433199 \)

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:  \(786647542210626533905079006002056634177=43^{8}\cdot 97^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $269.77$
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:  $43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{45529239285606077404461542612295654407921166133363016285599127691641} a^{15} + \frac{1666217159344915422587078958815479857758895915755158072536035690073}{3502249175815852108035503277868896492917012779489462791199932899357} a^{14} - \frac{2955847608875192852938934726536005914883383831443084979761176059541}{45529239285606077404461542612295654407921166133363016285599127691641} a^{13} - \frac{16585234943370514886678290099089321858150285903983872306795928840977}{45529239285606077404461542612295654407921166133363016285599127691641} a^{12} + \frac{15972549030607040769384278267159650892926592196391015016093838930237}{45529239285606077404461542612295654407921166133363016285599127691641} a^{11} - \frac{4920424879474543737811304981085620738110400584482047517505451609663}{45529239285606077404461542612295654407921166133363016285599127691641} a^{10} + \frac{14368272220351380411049245325499863400369738677783500629967033043206}{45529239285606077404461542612295654407921166133363016285599127691641} a^{9} - \frac{14199391681660297860908923165396561418250315559725480366731457595012}{45529239285606077404461542612295654407921166133363016285599127691641} a^{8} + \frac{862551700776192541159781630296101383145686684894424917270034038346}{4139021753236916127678322055663241309811015103033001480509011608331} a^{7} - \frac{12931570255149118198712695588464728760919591801551060131582115286176}{45529239285606077404461542612295654407921166133363016285599127691641} a^{6} - \frac{656372239217456655767974055130529899610126303049885807272727665998}{4139021753236916127678322055663241309811015103033001480509011608331} a^{5} - \frac{20667063770255555272862770328443106257169955027481063981352881324636}{45529239285606077404461542612295654407921166133363016285599127691641} a^{4} - \frac{15651370299064981815076977628426308510563493590080023022313641467865}{45529239285606077404461542612295654407921166133363016285599127691641} a^{3} - \frac{9798156305682228036580240819148771932219025820108866344237711703730}{45529239285606077404461542612295654407921166133363016285599127691641} a^{2} + \frac{139139950771179908564915916649700486601132739866035761586490410042}{45529239285606077404461542612295654407921166133363016285599127691641} a - \frac{15702492983688991330996333146204768437139151139569934909022786111378}{45529239285606077404461542612295654407921166133363016285599127691641}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}$, which has order $64$ (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:  \( -1 \) (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:  \( 66420577588.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-43}) \), 4.0.179353.1, 8.0.29358407457971857.3

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

Sibling fields

Degree 16 sibling: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.4.3.1$x^{4} - 97$$4$$1$$3$$C_4$$[\ ]_{4}$
97.4.3.1$x^{4} - 97$$4$$1$$3$$C_4$$[\ ]_{4}$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$