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Properties

Label 16.0.64005517659...0929.3
Degree $16$
Signature $[0, 8]$
Discriminant $29^{14}\cdot 173^{14}$
Root discriminant $1729.37$
Ramified primes $29, 173$
Class number $219869584$ (GRH)
Class group $[2, 7414, 14828]$ (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![11763847569050485421, -1427922201066519125, -8951931674854952, -16318805569604653, 2753263432891772, -130418478207074, 11615377868028, -1370337802129, 110959531595, -4141244433, 373695495, -13245362, 1465823, -31410, 1873, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 1873*x^14 - 31410*x^13 + 1465823*x^12 - 13245362*x^11 + 373695495*x^10 - 4141244433*x^9 + 110959531595*x^8 - 1370337802129*x^7 + 11615377868028*x^6 - 130418478207074*x^5 + 2753263432891772*x^4 - 16318805569604653*x^3 - 8951931674854952*x^2 - 1427922201066519125*x + 11763847569050485421)
 
gp: K = bnfinit(x^16 - 6*x^15 + 1873*x^14 - 31410*x^13 + 1465823*x^12 - 13245362*x^11 + 373695495*x^10 - 4141244433*x^9 + 110959531595*x^8 - 1370337802129*x^7 + 11615377868028*x^6 - 130418478207074*x^5 + 2753263432891772*x^4 - 16318805569604653*x^3 - 8951931674854952*x^2 - 1427922201066519125*x + 11763847569050485421, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 1873 x^{14} - 31410 x^{13} + 1465823 x^{12} - 13245362 x^{11} + 373695495 x^{10} - 4141244433 x^{9} + 110959531595 x^{8} - 1370337802129 x^{7} + 11615377868028 x^{6} - 130418478207074 x^{5} + 2753263432891772 x^{4} - 16318805569604653 x^{3} - 8951931674854952 x^{2} - 1427922201066519125 x + 11763847569050485421 \)

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:  \(6400551765976000463282291896137640731464336249990929=29^{14}\cdot 173^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1729.37$
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:  $29, 173$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{26} a^{12} - \frac{1}{26} a^{11} - \frac{5}{13} a^{10} - \frac{1}{2} a^{9} - \frac{6}{13} a^{8} - \frac{5}{26} a^{7} + \frac{1}{26} a^{6} - \frac{11}{26} a^{5} + \frac{3}{26} a^{4} - \frac{6}{13} a^{3} - \frac{4}{13} a^{2} + \frac{3}{13} a - \frac{9}{26}$, $\frac{1}{1794} a^{13} + \frac{3}{598} a^{12} - \frac{205}{897} a^{11} + \frac{823}{1794} a^{10} - \frac{188}{897} a^{9} + \frac{785}{1794} a^{8} - \frac{179}{1794} a^{7} + \frac{623}{1794} a^{6} + \frac{283}{1794} a^{5} + \frac{107}{299} a^{4} + \frac{40}{897} a^{3} + \frac{340}{897} a^{2} - \frac{157}{1794} a - \frac{11}{39}$, $\frac{1}{995680764} a^{14} + \frac{135139}{497840382} a^{13} - \frac{2443513}{497840382} a^{12} + \frac{5300575}{55315598} a^{11} - \frac{188259887}{995680764} a^{10} + \frac{21957955}{55315598} a^{9} + \frac{4077085}{38295414} a^{8} - \frac{449739005}{995680764} a^{7} + \frac{2004779}{76590828} a^{6} - \frac{97259195}{497840382} a^{5} - \frac{318332401}{995680764} a^{4} + \frac{7389677}{82973397} a^{3} - \frac{73118741}{331893588} a^{2} + \frac{425294663}{995680764} a + \frac{1572235}{43290468}$, $\frac{1}{270598162188150492893014391368155847117059098501476966109473581851452129298518564365288511001089423338607072567360751784} a^{15} + \frac{80266617657987581867166223878280050411859164268281700915597744189539076121583840691992246610386163767172919449}{270598162188150492893014391368155847117059098501476966109473581851452129298518564365288511001089423338607072567360751784} a^{14} + \frac{15071163950403216490231861779600525028821092697030660332834180759022652787462711648231981973690028801761411823731587}{67649540547037623223253597842038961779264774625369241527368395462863032324629641091322127750272355834651768141840187946} a^{13} + \frac{811025389395554497526717034949151290358139290047639785392045551737356600102550433464964947941630958488465739275034067}{135299081094075246446507195684077923558529549250738483054736790925726064649259282182644255500544711669303536283680375892} a^{12} + \frac{57510656654719545616394192831294925737042608530748903890078306947729531736052127656569262129778043932314059378321965701}{270598162188150492893014391368155847117059098501476966109473581851452129298518564365288511001089423338607072567360751784} a^{11} - \frac{49427810123594976004568718888627013488497810678743248063612957410564804103906377927740606977607315784536580786724896239}{270598162188150492893014391368155847117059098501476966109473581851452129298518564365288511001089423338607072567360751784} a^{10} + \frac{37513457123360603271594357941752655846813041486027181580588131809886322325366029182135909790070997555111790744373168891}{135299081094075246446507195684077923558529549250738483054736790925726064649259282182644255500544711669303536283680375892} a^{9} + \frac{35378474329776581207486805912412518451025489030078192917493987060792890258663559203282469718335327681950388395052332265}{270598162188150492893014391368155847117059098501476966109473581851452129298518564365288511001089423338607072567360751784} a^{8} + \frac{36605050806870078067219952796394219785977390646765098427887814228144855014237917413992557207344353507355077607367956923}{135299081094075246446507195684077923558529549250738483054736790925726064649259282182644255500544711669303536283680375892} a^{7} + \frac{37848131951820191917282849791620175530535380827559980445376336415527538459670025060299106051633781541508919279305922651}{90199387396050164297671463789385282372353032833825655369824527283817376432839521455096170333696474446202357522453583928} a^{6} + \frac{2447296449613849890112592047798705164947680819901150247310717663215805397058031142398029437088506324337140368237442617}{90199387396050164297671463789385282372353032833825655369824527283817376432839521455096170333696474446202357522453583928} a^{5} - \frac{57477714573466231922425112672724198973625091356095353382350420792811490199081166414488761693060909442294038235354968489}{270598162188150492893014391368155847117059098501476966109473581851452129298518564365288511001089423338607072567360751784} a^{4} + \frac{21998546905051698555365816106728289047817629603705148551975686575748193685104958592666412642811952123523585269319754539}{90199387396050164297671463789385282372353032833825655369824527283817376432839521455096170333696474446202357522453583928} a^{3} + \frac{452949856514256736665808415436974585305430700865815487919084615519043238317918330478872589826835544109878578044640775}{2019389270060824573828465607225043635201933570906544523205026730234717382824765405711108291052906144317963228114632476} a^{2} - \frac{1491874648769568375923931899959262318618332342702554679857767046083233579482821571829249156305983433832737543315772845}{5011077077558342460981747988299182354019612935212536409434695960212076468491084525283120574094248580344575417914087996} a + \frac{5368254690514964886205318336906714571960999025414493017143580516153743690854066518233464106094873076260111024921354479}{11765137486441325777957147450789384657263439065281607222151025297889223012979068015882109173960409710374220546406989208}$

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

Class group and class number

$C_{2}\times C_{7414}\times C_{14828}$, which has order $219869584$ (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:  \( 141448785521 \) (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

\(\Q(\sqrt{5017}) \), \(\Q(\sqrt{173}) \), \(\Q(\sqrt{29}) \), 4.4.126279339913.2, 4.4.126279339913.1, \(\Q(\sqrt{29}, \sqrt{173})\), 8.8.15946471688862994847569.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
$173$173.8.7.1$x^{8} - 173$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
173.8.7.1$x^{8} - 173$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$