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Properties

Label 16.0.41547610519...4673.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 97^{15}$
Root discriminant $126.23$
Ramified primes $3, 97$
Class number $32$ (GRH)
Class group $[32]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51840000, -76032000, 10833600, 12698880, 6436324, -3721270, -667219, 264213, 11734, 20935, -4493, -1274, 211, -45, 26, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 26*x^14 - 45*x^13 + 211*x^12 - 1274*x^11 - 4493*x^10 + 20935*x^9 + 11734*x^8 + 264213*x^7 - 667219*x^6 - 3721270*x^5 + 6436324*x^4 + 12698880*x^3 + 10833600*x^2 - 76032000*x + 51840000)
 
gp: K = bnfinit(x^16 - 7*x^15 + 26*x^14 - 45*x^13 + 211*x^12 - 1274*x^11 - 4493*x^10 + 20935*x^9 + 11734*x^8 + 264213*x^7 - 667219*x^6 - 3721270*x^5 + 6436324*x^4 + 12698880*x^3 + 10833600*x^2 - 76032000*x + 51840000, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 26 x^{14} - 45 x^{13} + 211 x^{12} - 1274 x^{11} - 4493 x^{10} + 20935 x^{9} + 11734 x^{8} + 264213 x^{7} - 667219 x^{6} - 3721270 x^{5} + 6436324 x^{4} + 12698880 x^{3} + 10833600 x^{2} - 76032000 x + 51840000 \)

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:  \(4154761051926475161829933538084673=3^{8}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.23$
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:  $3, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{60} a^{8} - \frac{1}{15} a^{7} + \frac{1}{15} a^{6} + \frac{1}{30} a^{5} - \frac{7}{30} a^{4} - \frac{1}{15} a^{3} + \frac{3}{20} a^{2} - \frac{3}{10} a$, $\frac{1}{180} a^{9} - \frac{1}{15} a^{7} - \frac{1}{15} a^{6} - \frac{1}{30} a^{5} - \frac{1}{6} a^{4} + \frac{83}{180} a^{3} + \frac{13}{30} a^{2} + \frac{13}{30} a$, $\frac{1}{360} a^{10} - \frac{1}{360} a^{9} + \frac{1}{30} a^{7} + \frac{1}{15} a^{6} + \frac{1}{6} a^{5} - \frac{17}{72} a^{4} + \frac{97}{360} a^{3} - \frac{1}{30} a^{2} - \frac{1}{15} a$, $\frac{1}{1080} a^{11} - \frac{1}{1080} a^{10} + \frac{1}{180} a^{8} - \frac{1}{15} a^{7} + \frac{1}{30} a^{6} - \frac{217}{1080} a^{5} + \frac{181}{1080} a^{4} + \frac{7}{30} a^{3} - \frac{73}{180} a^{2} + \frac{13}{30} a$, $\frac{1}{2160} a^{12} + \frac{1}{1080} a^{10} + \frac{1}{720} a^{9} + \frac{1}{360} a^{8} + \frac{1}{30} a^{7} + \frac{179}{2160} a^{6} + \frac{1}{20} a^{5} + \frac{25}{216} a^{4} - \frac{301}{720} a^{3} - \frac{163}{360} a^{2} - \frac{5}{12} a$, $\frac{1}{8812800} a^{13} + \frac{643}{4406400} a^{12} + \frac{521}{2203200} a^{11} - \frac{2273}{8812800} a^{10} - \frac{23}{1468800} a^{9} - \frac{2377}{367200} a^{8} + \frac{241523}{8812800} a^{7} + \frac{135887}{4406400} a^{6} + \frac{1141}{550800} a^{5} - \frac{132259}{8812800} a^{4} + \frac{729569}{1468800} a^{3} - \frac{214069}{734400} a^{2} + \frac{1057}{4080} a - \frac{31}{68}$, $\frac{1}{317260800} a^{14} - \frac{1}{158630400} a^{13} - \frac{497}{26438400} a^{12} + \frac{2899}{63452160} a^{11} - \frac{74417}{158630400} a^{10} + \frac{15229}{13219200} a^{9} - \frac{2266573}{317260800} a^{8} + \frac{1937639}{31726080} a^{7} - \frac{104501}{3304800} a^{6} - \frac{36334787}{317260800} a^{5} + \frac{2829983}{158630400} a^{4} - \frac{2396533}{5287680} a^{3} - \frac{333257}{6609600} a^{2} + \frac{16181}{36720} a - \frac{109}{306}$, $\frac{1}{1439382469398596068199040000} a^{15} + \frac{13298251859758307}{26655230914788816077760000} a^{14} + \frac{3293490785489347357}{179922808674824508524880000} a^{13} + \frac{19677571767168458087083}{287876493879719213639808000} a^{12} + \frac{36692499033201171160877}{79965692744366448233280000} a^{11} - \frac{422938182688967511125891}{359845617349649017049760000} a^{10} - \frac{3549672994734181176974533}{1439382469398596068199040000} a^{9} - \frac{35225468528366060643449}{47979415646619868939968000} a^{8} + \frac{2167095378431788582292971}{359845617349649017049760000} a^{7} - \frac{24064738555094322623340347}{1439382469398596068199040000} a^{6} - \frac{1094545786691905296479293}{7269608431306040748480000} a^{5} + \frac{2893634001065625188522831}{35984561734964901704976000} a^{4} + \frac{793874650971717081767947}{1817402107826510187120000} a^{3} + \frac{916622770805952336372193}{5997426955827483617496000} a^{2} - \frac{4981361075828893863001}{33319038643486020097200} a + \frac{185845794026820405463}{555317310724767001620}$

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

Class group and class number

$C_{32}$, which has order $32$ (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:  \( 150480538543 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

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 $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-291}) \), 4.0.8214057.2, 8.0.6544661042727153.2

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
97Data not computed