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Properties

Label 16.0.80341646997...4929.7
Degree $16$
Signature $[0, 8]$
Discriminant $37^{12}\cdot 73^{14}$
Root discriminant $640.56$
Ramified primes $37, 73$
Class number $2624$ (GRH)
Class group $[2, 2, 2, 328]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![62802119569213, 14259307873786, 4606618252806, 91847336586, -11184167097, -9149326000, -797038879, 205373656, -13540961, 3506890, 286426, -93666, 16630, 896, -251, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 251*x^14 + 896*x^13 + 16630*x^12 - 93666*x^11 + 286426*x^10 + 3506890*x^9 - 13540961*x^8 + 205373656*x^7 - 797038879*x^6 - 9149326000*x^5 - 11184167097*x^4 + 91847336586*x^3 + 4606618252806*x^2 + 14259307873786*x + 62802119569213)
 
gp: K = bnfinit(x^16 - 4*x^15 - 251*x^14 + 896*x^13 + 16630*x^12 - 93666*x^11 + 286426*x^10 + 3506890*x^9 - 13540961*x^8 + 205373656*x^7 - 797038879*x^6 - 9149326000*x^5 - 11184167097*x^4 + 91847336586*x^3 + 4606618252806*x^2 + 14259307873786*x + 62802119569213, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 251 x^{14} + 896 x^{13} + 16630 x^{12} - 93666 x^{11} + 286426 x^{10} + 3506890 x^{9} - 13540961 x^{8} + 205373656 x^{7} - 797038879 x^{6} - 9149326000 x^{5} - 11184167097 x^{4} + 91847336586 x^{3} + 4606618252806 x^{2} + 14259307873786 x + 62802119569213 \)

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:  \(803416469975725073264940954221144185207724929=37^{12}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $640.56$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{74} a^{8} - \frac{1}{37} a^{7} + \frac{1}{37} a^{6} + \frac{4}{37} a^{5} - \frac{33}{74} a^{4} - \frac{18}{37} a^{3} - \frac{15}{74} a^{2} + \frac{25}{74} a + \frac{5}{37}$, $\frac{1}{74} a^{9} - \frac{1}{37} a^{7} + \frac{6}{37} a^{6} - \frac{17}{74} a^{5} - \frac{14}{37} a^{4} - \frac{13}{74} a^{3} - \frac{5}{74} a^{2} - \frac{7}{37} a + \frac{10}{37}$, $\frac{1}{74} a^{10} + \frac{4}{37} a^{7} - \frac{13}{74} a^{6} - \frac{6}{37} a^{5} - \frac{5}{74} a^{4} - \frac{3}{74} a^{3} + \frac{15}{37} a^{2} - \frac{2}{37} a + \frac{10}{37}$, $\frac{1}{74} a^{11} + \frac{3}{74} a^{7} - \frac{14}{37} a^{6} + \frac{5}{74} a^{5} - \frac{35}{74} a^{4} + \frac{11}{37} a^{3} - \frac{16}{37} a^{2} - \frac{16}{37} a - \frac{3}{37}$, $\frac{1}{222} a^{12} + \frac{1}{222} a^{11} + \frac{1}{222} a^{9} + \frac{8}{111} a^{7} - \frac{91}{222} a^{6} + \frac{1}{74} a^{5} - \frac{15}{37} a^{4} + \frac{8}{37} a^{3} - \frac{49}{111} a^{2} + \frac{29}{111} a + \frac{95}{222}$, $\frac{1}{222} a^{13} - \frac{1}{222} a^{11} + \frac{1}{222} a^{10} - \frac{1}{222} a^{9} + \frac{1}{222} a^{8} + \frac{17}{111} a^{7} + \frac{32}{111} a^{6} + \frac{3}{74} a^{5} - \frac{11}{74} a^{4} + \frac{61}{222} a^{3} - \frac{21}{74} a^{2} - \frac{5}{222} a + \frac{44}{111}$, $\frac{1}{54296389704} a^{14} - \frac{2712153}{4524699142} a^{13} + \frac{12083437}{27148194852} a^{12} + \frac{474457}{366867498} a^{11} - \frac{15658889}{6787048713} a^{10} + \frac{99490565}{27148194852} a^{9} - \frac{153920443}{27148194852} a^{8} - \frac{1360455940}{6787048713} a^{7} - \frac{29269557}{18098796568} a^{6} - \frac{716610237}{4524699142} a^{5} + \frac{2237280251}{27148194852} a^{4} - \frac{1118608845}{2262349571} a^{3} - \frac{22727193323}{54296389704} a^{2} + \frac{382156769}{27148194852} a + \frac{2834743041}{18098796568}$, $\frac{1}{8401503432935691167337782621290875109195320771798158227762539370424496} a^{15} + \frac{57233855170940873703313660896825653806947096248021156533297}{8401503432935691167337782621290875109195320771798158227762539370424496} a^{14} + \frac{3602330584964228327550953630888341297599829379512215307181213310655}{4200751716467845583668891310645437554597660385899079113881269685212248} a^{13} - \frac{3122266035787866672421635635931433975224712075526198163537441058447}{1400250572155948527889630436881812518199220128633026371293756561737416} a^{12} - \frac{1750231828891284195591649644066006685283440393193710067769543925853}{700125286077974263944815218440906259099610064316513185646878280868708} a^{11} + \frac{3513998389830209337103144674696495643234162535878739943108963505203}{1400250572155948527889630436881812518199220128633026371293756561737416} a^{10} + \frac{1068224436295546557306667073622101186549221646982408563073621373691}{2100375858233922791834445655322718777298830192949539556940634842606124} a^{9} + \frac{15657507904198888590519687301146549195075532775225633689607864560339}{4200751716467845583668891310645437554597660385899079113881269685212248} a^{8} - \frac{1931785928073884223861810548073437327298702808853959335471467539122371}{8401503432935691167337782621290875109195320771798158227762539370424496} a^{7} + \frac{145927151114311701558599303219754146154275040891990286489538546616043}{2800501144311897055779260873763625036398440257266052742587513123474832} a^{6} + \frac{1731706624226267470463909722941081657383276620958831726477240617823357}{4200751716467845583668891310645437554597660385899079113881269685212248} a^{5} - \frac{49119655500387345089435823136489311748330049135856877549674880137955}{113533830174806637396456521909336150124261091510785921996791072573304} a^{4} + \frac{2014485666854541798180976721702857354468415732676928445282982591766657}{8401503432935691167337782621290875109195320771798158227762539370424496} a^{3} + \frac{174351623549836765336582391296445260101547296125215038676268738315975}{8401503432935691167337782621290875109195320771798158227762539370424496} a^{2} + \frac{164493924551872540635697580908562090043337398530652956993023744621285}{8401503432935691167337782621290875109195320771798158227762539370424496} a - \frac{841792445069194674787627229040740912344602714217423021734582742359951}{2800501144311897055779260873763625036398440257266052742587513123474832}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{328}$, which has order $2624$ (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:  \( 1747700266780 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.0.20704603455949352617.2

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

Sibling fields

Degree 16 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

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
37Data not computed
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$