Properties

Label 16.16.706...153.1
Degree $16$
Signature $[16, 0]$
Discriminant $7.070\times 10^{41}$
Root discriminant $412.66$
Ramified primes $17, 89$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 569*x^14 + 3985*x^13 + 110105*x^12 - 1255302*x^11 - 6017305*x^10 + 141442454*x^9 - 386991894*x^8 - 4380441354*x^7 + 34047900007*x^6 - 66542719513*x^5 - 128137329112*x^4 + 622983352474*x^3 - 407875925600*x^2 - 585981669666*x + 146245588969)
 
gp: K = bnfinit(x^16 - 3*x^15 - 569*x^14 + 3985*x^13 + 110105*x^12 - 1255302*x^11 - 6017305*x^10 + 141442454*x^9 - 386991894*x^8 - 4380441354*x^7 + 34047900007*x^6 - 66542719513*x^5 - 128137329112*x^4 + 622983352474*x^3 - 407875925600*x^2 - 585981669666*x + 146245588969, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![146245588969, -585981669666, -407875925600, 622983352474, -128137329112, -66542719513, 34047900007, -4380441354, -386991894, 141442454, -6017305, -1255302, 110105, 3985, -569, -3, 1]);
 

\( x^{16} - 3 x^{15} - 569 x^{14} + 3985 x^{13} + 110105 x^{12} - 1255302 x^{11} - 6017305 x^{10} + 141442454 x^{9} - 386991894 x^{8} - 4380441354 x^{7} + 34047900007 x^{6} - 66542719513 x^{5} - 128137329112 x^{4} + 622983352474 x^{3} - 407875925600 x^{2} - 585981669666 x + 146245588969 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[16, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(706991024666918541719408792174287819307153\)\(\medspace = 17^{15}\cdot 89^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $412.66$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $17, 89$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{67} a^{13} + \frac{26}{67} a^{12} + \frac{23}{67} a^{11} - \frac{7}{67} a^{10} + \frac{17}{67} a^{9} - \frac{1}{67} a^{8} - \frac{24}{67} a^{7} + \frac{1}{67} a^{6} - \frac{19}{67} a^{5} - \frac{15}{67} a^{4} - \frac{12}{67} a^{3} - \frac{18}{67} a^{2} - \frac{9}{67} a - \frac{11}{67}$, $\frac{1}{871} a^{14} - \frac{5}{871} a^{13} + \frac{21}{871} a^{12} + \frac{419}{871} a^{11} - \frac{168}{871} a^{10} - \frac{20}{67} a^{9} + \frac{16}{67} a^{8} + \frac{75}{871} a^{7} + \frac{419}{871} a^{6} - \frac{163}{871} a^{5} + \frac{185}{871} a^{4} + \frac{19}{871} a^{3} + \frac{415}{871} a^{2} + \frac{408}{871}$, $\frac{1}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{15} + \frac{35352135003902983617501052017030028192830541458140697352790747022376}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{14} - \frac{664614045716794451415052635840725218760678719746805995739383639174125}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{13} + \frac{22231374688266891990400625013318590832303034084007006646283014707050794}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{12} - \frac{58513323326256711736101365078630475418174379904987935366997025805234909}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{11} - \frac{57962051316850517711147930506165491108207952313944195267209631511819640}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{10} - \frac{2873222762284371235555062147703715372389391438300734968669361382525588}{10493966348975693273292552081704336914644924502500166846341738406041839} a^{9} - \frac{64329685419849606906309204284662094601526188070912255907967640420857664}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{8} - \frac{33573697902735023003545555662454488895470452267132122698465841119190272}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{7} - \frac{33217421321806907647433404224482001925088488065538122182477777260744733}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{6} - \frac{26972324098493518686588751834503794695269733928264812435380453252534893}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{5} - \frac{66444426441775768807106152708218748041965924700616523542317229360866259}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{4} - \frac{63528390429130089315022455555222951157271087335913461950936197360581825}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{3} + \frac{20244200382538588245697311019658264016231286164334350707144497685628995}{136421562536684012552803177062156379890384018532502169002442599278543907} a^{2} + \frac{58701369274580027398718380017257702543057289260666801676810620031535250}{136421562536684012552803177062156379890384018532502169002442599278543907} a + \frac{29033643965474469063735702738424240205689316968800575177228422799803}{307949351098609509148539903074845101332695301427770133188358011915449}$

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

Class group and class number

$C_{4}$, which has order $4$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2175501728390000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{16}\cdot(2\pi)^{0}\cdot 2175501728390000 \cdot 4}{2\sqrt{706991024666918541719408792174287819307153}}\approx 0.339127031797577$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.25745567912986193.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
17Data not computed
$89$89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$