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Properties

Label 16.16.7910059100...0881.1
Degree $16$
Signature $[16, 0]$
Discriminant $29^{14}\cdot 149^{14}$
Root discriminant $1517.53$
Ramified primes $29, 149$
Class number $20$ (GRH)
Class group $[20]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79952241592615927, -90854176564154546, 20698040251059424, 4398309541608152, -2133469310847068, 183380655430714, 20267517960617, -3724774186692, 34138973916, 20629844262, -785911195, -43338060, 2418348, 33440, -2706, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 2706*x^14 + 33440*x^13 + 2418348*x^12 - 43338060*x^11 - 785911195*x^10 + 20629844262*x^9 + 34138973916*x^8 - 3724774186692*x^7 + 20267517960617*x^6 + 183380655430714*x^5 - 2133469310847068*x^4 + 4398309541608152*x^3 + 20698040251059424*x^2 - 90854176564154546*x + 79952241592615927)
 
gp: K = bnfinit(x^16 - 6*x^15 - 2706*x^14 + 33440*x^13 + 2418348*x^12 - 43338060*x^11 - 785911195*x^10 + 20629844262*x^9 + 34138973916*x^8 - 3724774186692*x^7 + 20267517960617*x^6 + 183380655430714*x^5 - 2133469310847068*x^4 + 4398309541608152*x^3 + 20698040251059424*x^2 - 90854176564154546*x + 79952241592615927, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 2706 x^{14} + 33440 x^{13} + 2418348 x^{12} - 43338060 x^{11} - 785911195 x^{10} + 20629844262 x^{9} + 34138973916 x^{8} - 3724774186692 x^{7} + 20267517960617 x^{6} + 183380655430714 x^{5} - 2133469310847068 x^{4} + 4398309541608152 x^{3} + 20698040251059424 x^{2} - 90854176564154546 x + 79952241592615927 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(791005910030165719912925444172568170567213001250881=29^{14}\cdot 149^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1517.53$
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:  $29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{10} a^{6} + \frac{3}{10} a^{5} - \frac{1}{10} a^{3} - \frac{1}{10} a^{2} + \frac{3}{10} a - \frac{3}{10}$, $\frac{1}{864100} a^{14} - \frac{8079}{432050} a^{13} - \frac{201541}{864100} a^{12} + \frac{8573}{86410} a^{11} - \frac{25171}{864100} a^{10} + \frac{16251}{432050} a^{9} - \frac{11939}{432050} a^{8} + \frac{107933}{432050} a^{7} + \frac{100703}{216025} a^{6} + \frac{62117}{216025} a^{5} + \frac{129669}{864100} a^{4} - \frac{83548}{216025} a^{3} - \frac{416053}{864100} a^{2} + \frac{5627}{17282} a + \frac{28627}{864100}$, $\frac{1}{1293942057494124514491259457487980041862347612480752391391978739444495985910728397330614118226596037914873000} a^{15} + \frac{55448444049475115948796414496571454470165648455794257271060750590085371267364844499986258038580972973}{1293942057494124514491259457487980041862347612480752391391978739444495985910728397330614118226596037914873000} a^{14} - \frac{39599932449031322928297363312631343052765818321431229307349408492235278564062912573359265382806350821303039}{1293942057494124514491259457487980041862347612480752391391978739444495985910728397330614118226596037914873000} a^{13} - \frac{116940683941072952590294188657812246497576430209197732215574694291879252659578477911641315695137098696604041}{1293942057494124514491259457487980041862347612480752391391978739444495985910728397330614118226596037914873000} a^{12} + \frac{101449886913890860311830554118225770086623522726322936618138543908235499215698921964247468931187153612497109}{1293942057494124514491259457487980041862347612480752391391978739444495985910728397330614118226596037914873000} a^{11} + \frac{198951516838030888164764817301096081429886729492706908194458817118126458599235221122445314004302338510386651}{1293942057494124514491259457487980041862347612480752391391978739444495985910728397330614118226596037914873000} a^{10} - \frac{49146724240383864231864443893745821673781762831639302199796336416782723560354712175898834284458758903417483}{646971028747062257245629728743990020931173806240376195695989369722247992955364198665307059113298018957436500} a^{9} - \frac{36817794139890631678856438749423556196091114464115401284427321220819763310922452464718786967986674655651719}{161742757186765564311407432185997505232793451560094048923997342430561998238841049666326764778324504739359125} a^{8} + \frac{17214450382992832719931670420600361662762035831321642855520154314552541127886333647744394773883244317034227}{323485514373531128622814864371995010465586903120188097847994684861123996477682099332653529556649009478718250} a^{7} - \frac{16290897629734717053769650462294376840699279064038294235170718232251788328464949392929191732535526344837533}{64697102874706225724562972874399002093117380624037619569598936972224799295536419866530705911329801895743650} a^{6} - \frac{75421093567043520166844434684370200736526656860322917086222002364614424297963702522913512383007357047256823}{1293942057494124514491259457487980041862347612480752391391978739444495985910728397330614118226596037914873000} a^{5} + \frac{369469741489404495086148451775930357544182719107630619522802996248284858998029652368545284028964018544126297}{1293942057494124514491259457487980041862347612480752391391978739444495985910728397330614118226596037914873000} a^{4} + \frac{100625132039540734856825245693496872162081414509931792447289280249000426795477952810947813670049467072373219}{258788411498824902898251891497596008372469522496150478278395747888899197182145679466122823645319207582974600} a^{3} - \frac{567208830229830064853496148715166199003888739992705653068271535103740288360187660622417618461911653464040543}{1293942057494124514491259457487980041862347612480752391391978739444495985910728397330614118226596037914873000} a^{2} + \frac{167000399224522255687760624737207572045821457716008633375043613887839241964846475445566614602664180939310027}{1293942057494124514491259457487980041862347612480752391391978739444495985910728397330614118226596037914873000} a + \frac{281197324757498257634642083157766628015544155967432909385885808193232924370261914473786747952857366410681387}{1293942057494124514491259457487980041862347612480752391391978739444495985910728397330614118226596037914873000}$

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

Class group and class number

$C_{20}$, which has order $20$ (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:  $15$
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:  \( 7091423444250000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{4321}) \), \(\Q(\sqrt{149}) \), \(\Q(\sqrt{29}) \), 4.4.80677568161.2, 4.4.80677568161.1, \(\Q(\sqrt{29}, \sqrt{149})\), 8.8.6508870004372800921921.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$29$29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$149$149.8.7.1$x^{8} - 149$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
149.8.7.1$x^{8} - 149$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$