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Properties

Label 16.0.98884277765...0681.4
Degree $16$
Signature $[0, 8]$
Discriminant $41^{13}\cdot 83^{12}$
Root discriminant $561.95$
Ramified primes $41, 83$
Class number $48$ (GRH)
Class group $[4, 12]$ (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![80918703349713, -45824508231069, 13325887212862, -92057160074, -859514035076, 175446845335, 9233730823, -9810178344, 1211493195, 60514795, -15460825, -121864, 46435, 2645, 5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 5*x^14 + 2645*x^13 + 46435*x^12 - 121864*x^11 - 15460825*x^10 + 60514795*x^9 + 1211493195*x^8 - 9810178344*x^7 + 9233730823*x^6 + 175446845335*x^5 - 859514035076*x^4 - 92057160074*x^3 + 13325887212862*x^2 - 45824508231069*x + 80918703349713)
 
gp: K = bnfinit(x^16 - 4*x^15 + 5*x^14 + 2645*x^13 + 46435*x^12 - 121864*x^11 - 15460825*x^10 + 60514795*x^9 + 1211493195*x^8 - 9810178344*x^7 + 9233730823*x^6 + 175446845335*x^5 - 859514035076*x^4 - 92057160074*x^3 + 13325887212862*x^2 - 45824508231069*x + 80918703349713, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 5 x^{14} + 2645 x^{13} + 46435 x^{12} - 121864 x^{11} - 15460825 x^{10} + 60514795 x^{9} + 1211493195 x^{8} - 9810178344 x^{7} + 9233730823 x^{6} + 175446845335 x^{5} - 859514035076 x^{4} - 92057160074 x^{3} + 13325887212862 x^{2} - 45824508231069 x + 80918703349713 \)

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:  \(98884277765511220794553278305197522019930681=41^{13}\cdot 83^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $561.95$
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:  $41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} + \frac{4}{9} a^{11} - \frac{1}{9} a^{10} + \frac{4}{9} a^{9} + \frac{4}{9} a^{7} - \frac{2}{9} a^{6} + \frac{4}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{24950374790776867696199539931071756540649846562003110803018975624817041210439308907394496826301588687} a^{15} + \frac{414996956099362478006083128358563993410737613556103063336300842897755251161910305545133254707957522}{8316791596925622565399846643690585513549948854001036934339658541605680403479769635798165608767196229} a^{14} - \frac{1130655234079398591756194687031652267056837155958058049555671783595383406089030077785547591216218827}{24950374790776867696199539931071756540649846562003110803018975624817041210439308907394496826301588687} a^{13} + \frac{6043755807259337420441307922587671496963149592293357882543915561150427115379890693148602187224014088}{24950374790776867696199539931071756540649846562003110803018975624817041210439308907394496826301588687} a^{12} + \frac{12339901001960765498039978969552482966070343496444615018777839507035189054426833754120286101458292849}{24950374790776867696199539931071756540649846562003110803018975624817041210439308907394496826301588687} a^{11} - \frac{7895258395981454942713107095948715634105441768131953468934338437814568550447845529080902391859383755}{24950374790776867696199539931071756540649846562003110803018975624817041210439308907394496826301588687} a^{10} + \frac{2646961422127275157502288997866477785020404171886989936793224187993007524046882747774964785160836525}{8316791596925622565399846643690585513549948854001036934339658541605680403479769635798165608767196229} a^{9} - \frac{8843865159841834171268485336284990202746088301647327299260628698887741659420301672946186403320048699}{24950374790776867696199539931071756540649846562003110803018975624817041210439308907394496826301588687} a^{8} + \frac{5429833436658740986021358119362133878114019311435633848319236764858485003534478462381950574474712476}{24950374790776867696199539931071756540649846562003110803018975624817041210439308907394496826301588687} a^{7} + \frac{79042139982143747021389764421198968063616537742340822344531165824175648497920087351771205450464679}{24950374790776867696199539931071756540649846562003110803018975624817041210439308907394496826301588687} a^{6} + \frac{12160052552784860357692296431790071073509216569732701625611349295594456993691802319382447518184121595}{24950374790776867696199539931071756540649846562003110803018975624817041210439308907394496826301588687} a^{5} + \frac{2304344491278020231086818902721831523429690374424678196567369606702035793464679595314358504931207040}{8316791596925622565399846643690585513549948854001036934339658541605680403479769635798165608767196229} a^{4} + \frac{1475572707961546302106738014263319810646817149639483675098994547488514319895201438656335272594138843}{24950374790776867696199539931071756540649846562003110803018975624817041210439308907394496826301588687} a^{3} - \frac{2865861272496395517853552624326924689496217292602019840806520296238294907153437083975116951142951834}{24950374790776867696199539931071756540649846562003110803018975624817041210439308907394496826301588687} a^{2} - \frac{1927611596376111754824095226176685184957381715294015946191721179544526211572951574728394307175062451}{8316791596925622565399846643690585513549948854001036934339658541605680403479769635798165608767196229} a - \frac{287089964718464534936494759164456624754059466122545002629969860622295113342335120373269331834558697}{924087955213958062822205182632287279283327650444559659371073171289520044831085515088685067640799581}$

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

Class group and class number

$C_{4}\times C_{12}$, which has order $48$ (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:  \( 185460517881000 \) (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

\(\Q(\sqrt{-83}) \), 4.0.282449.1, 8.0.5498340776898521.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$41$41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.8.7.2$x^{8} - 1476$$8$$1$$7$$C_8$$[\ ]_{8}$
$83$83.8.6.2$x^{8} + 249 x^{4} + 27556$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
83.8.6.2$x^{8} + 249 x^{4} + 27556$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$