↑

Properties

Label 16.8.80291031759...9473.2
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 73^{15}$
Root discriminant $1315.35$
Ramified primes $37, 73$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![167287637065831, 37731456819936, -53609796972279, -32893267183391, -3924866163908, 232338128224, -53629386313, -3023072037, 2625021761, 3398924, 8593, -46787, -107320, 119, -29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 119*x^13 - 107320*x^12 - 46787*x^11 + 8593*x^10 + 3398924*x^9 + 2625021761*x^8 - 3023072037*x^7 - 53629386313*x^6 + 232338128224*x^5 - 3924866163908*x^4 - 32893267183391*x^3 - 53609796972279*x^2 + 37731456819936*x + 167287637065831)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 + 119*x^13 - 107320*x^12 - 46787*x^11 + 8593*x^10 + 3398924*x^9 + 2625021761*x^8 - 3023072037*x^7 - 53629386313*x^6 + 232338128224*x^5 - 3924866163908*x^4 - 32893267183391*x^3 - 53609796972279*x^2 + 37731456819936*x + 167287637065831, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 29 x^{14} + 119 x^{13} - 107320 x^{12} - 46787 x^{11} + 8593 x^{10} + 3398924 x^{9} + 2625021761 x^{8} - 3023072037 x^{7} - 53629386313 x^{6} + 232338128224 x^{5} - 3924866163908 x^{4} - 32893267183391 x^{3} - 53609796972279 x^{2} + 37731456819936 x + 167287637065831 \)

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:  \(80291031759964036646878404141998486437104406229473=37^{14}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1315.35$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{71017171836890150306} a^{14} + \frac{7632635414447169233}{71017171836890150306} a^{13} - \frac{9184134568998694287}{71017171836890150306} a^{12} - \frac{32313514283711794751}{71017171836890150306} a^{11} - \frac{15718322145611562431}{71017171836890150306} a^{10} - \frac{4981672232783337943}{71017171836890150306} a^{9} + \frac{93083209263417246}{35508585918445075153} a^{8} + \frac{539793392875536063}{1164215931752297546} a^{7} + \frac{2829024127096737337}{71017171836890150306} a^{6} + \frac{19246211657418681781}{71017171836890150306} a^{5} + \frac{26520225606566479739}{71017171836890150306} a^{4} + \frac{27609190371234195691}{71017171836890150306} a^{3} + \frac{4385884288866491041}{35508585918445075153} a^{2} + \frac{30110836892409507797}{71017171836890150306} a + \frac{391306085149197708}{1868872943076056587}$, $\frac{1}{1152872945950047515366348673124809226943363345348961019016901328220544713044602924934834} a^{15} + \frac{2880787794651353782353290292501329401809803856352931479006001300339}{576436472975023757683174336562404613471681672674480509508450664110272356522301462467417} a^{14} - \frac{64734407507626263863437888233034345783458638661597475064822057920745973946724258307804}{576436472975023757683174336562404613471681672674480509508450664110272356522301462467417} a^{13} - \frac{53946822394319386332540868487902658373065281703314096922786790327271215055868092924056}{576436472975023757683174336562404613471681672674480509508450664110272356522301462467417} a^{12} - \frac{63360020293200444700144441096226305725017749826913537645061494346876302408402210779636}{576436472975023757683174336562404613471681672674480509508450664110272356522301462467417} a^{11} - \frac{110496445107537096654609204770611699671448468612647433829877484807921832350401810701248}{576436472975023757683174336562404613471681672674480509508450664110272356522301462467417} a^{10} - \frac{457277852574588835372074296114679795284900639871350160694506315250346310263077557584519}{1152872945950047515366348673124809226943363345348961019016901328220544713044602924934834} a^{9} + \frac{389990096191123764692221103559065954557785558520548695298719238445698014245933828826845}{1152872945950047515366348673124809226943363345348961019016901328220544713044602924934834} a^{8} + \frac{112958259516892179300418541372568271332355813208431511507660137672177200657803940028554}{576436472975023757683174336562404613471681672674480509508450664110272356522301462467417} a^{7} - \frac{139437109771051870645321588687686269044828946294886801687449613747395637495290564183725}{576436472975023757683174336562404613471681672674480509508450664110272356522301462467417} a^{6} - \frac{1788050638681923051992650947703586644526345993490011883554234197131309166663537573281}{30338761735527566193851280871705505972193772246025289974128982321593281922226392761443} a^{5} + \frac{1374566501488927895114106131300905478412155221487386693743737611981493805547901638992}{7296664214873718451685751095726640676856730033854183664663932457092055145851917246423} a^{4} - \frac{499602070203154867256343076479111094471550137351292030864462131445434329025366579435941}{1152872945950047515366348673124809226943363345348961019016901328220544713044602924934834} a^{3} + \frac{284115832514514630747170375878734394740008287458324764412139338231676382342386849069213}{1152872945950047515366348673124809226943363345348961019016901328220544713044602924934834} a^{2} + \frac{248381581108849280193557187186686319236380479393234586212448108794643199790533296357935}{1152872945950047515366348673124809226943363345348961019016901328220544713044602924934834} a - \frac{3984514155444045445928404483761294024399538480076234014728427230248350583931830891400}{30338761735527566193851280871705505972193772246025289974128982321593281922226392761443}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 1801777968080000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.532564273.1, 8.8.28344602131194663732673.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $16$ $16$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$

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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
73Data not computed