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Properties

Label 16.0.68372792983...489.26
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 67^{8}$
Root discriminant $97.65$
Ramified primes $17, 67$
Class number $2304$ (GRH)
Class group $[2, 4, 12, 24]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1651813193, 1453238270, 1971691906, 1738422858, 574526983, -22418086, -40985220, -4397396, -212804, -30432, 46774, 5860, 434, 150, -58, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 58*x^14 + 150*x^13 + 434*x^12 + 5860*x^11 + 46774*x^10 - 30432*x^9 - 212804*x^8 - 4397396*x^7 - 40985220*x^6 - 22418086*x^5 + 574526983*x^4 + 1738422858*x^3 + 1971691906*x^2 + 1453238270*x + 1651813193)
 
gp: K = bnfinit(x^16 - 6*x^15 - 58*x^14 + 150*x^13 + 434*x^12 + 5860*x^11 + 46774*x^10 - 30432*x^9 - 212804*x^8 - 4397396*x^7 - 40985220*x^6 - 22418086*x^5 + 574526983*x^4 + 1738422858*x^3 + 1971691906*x^2 + 1453238270*x + 1651813193, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 58 x^{14} + 150 x^{13} + 434 x^{12} + 5860 x^{11} + 46774 x^{10} - 30432 x^{9} - 212804 x^{8} - 4397396 x^{7} - 40985220 x^{6} - 22418086 x^{5} + 574526983 x^{4} + 1738422858 x^{3} + 1971691906 x^{2} + 1453238270 x + 1651813193 \)

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:  \(68372792983010839504523425519489=17^{14}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.65$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{104} a^{12} + \frac{1}{26} a^{11} + \frac{1}{52} a^{10} - \frac{11}{104} a^{9} + \frac{7}{104} a^{8} - \frac{5}{26} a^{7} - \frac{1}{104} a^{6} + \frac{11}{52} a^{5} + \frac{17}{104} a^{4} - \frac{41}{104} a^{3} - \frac{15}{104} a^{2} + \frac{33}{104} a + \frac{41}{104}$, $\frac{1}{104} a^{13} + \frac{3}{26} a^{11} + \frac{7}{104} a^{10} - \frac{1}{104} a^{9} + \frac{1}{26} a^{8} - \frac{25}{104} a^{7} - \frac{1}{4} a^{6} + \frac{7}{104} a^{5} + \frac{21}{104} a^{4} - \frac{33}{104} a^{3} - \frac{37}{104} a^{2} - \frac{3}{8} a - \frac{1}{13}$, $\frac{1}{104} a^{14} + \frac{11}{104} a^{11} + \frac{1}{104} a^{10} + \frac{3}{52} a^{9} - \frac{5}{104} a^{8} + \frac{3}{52} a^{7} + \frac{19}{104} a^{6} + \frac{17}{104} a^{5} - \frac{3}{104} a^{4} - \frac{3}{8} a^{3} - \frac{41}{104} a^{2} - \frac{7}{52} a - \frac{3}{13}$, $\frac{1}{100375850547163313566098818489329089440692663904105484501361064} a^{15} + \frac{354094663183938372676774360143133263104287800997914040333167}{100375850547163313566098818489329089440692663904105484501361064} a^{14} - \frac{336929056850018271259343011998797692712646631035921968564339}{100375850547163313566098818489329089440692663904105484501361064} a^{13} + \frac{233590905985311281413081997408256873072273094430068381405111}{100375850547163313566098818489329089440692663904105484501361064} a^{12} + \frac{558885179272254254263333446258200395834703657791183747349231}{25093962636790828391524704622332272360173165976026371125340266} a^{11} - \frac{963762836063777656607537476543570967652614470260230100425630}{12546981318395414195762352311166136180086582988013185562670133} a^{10} - \frac{5921839919236327366450348473096127807652862228255740754992859}{50187925273581656783049409244664544720346331952052742250680532} a^{9} - \frac{109509038591022516730640730286745384221809562724175797104729}{100375850547163313566098818489329089440692663904105484501361064} a^{8} + \frac{772637809508490165144075476054120627944939031364206354732129}{12546981318395414195762352311166136180086582988013185562670133} a^{7} - \frac{24533823085380022376254023480761759002777829125125265774563}{25093962636790828391524704622332272360173165976026371125340266} a^{6} + \frac{540403458693212593035793341390100120751736925920648747888165}{100375850547163313566098818489329089440692663904105484501361064} a^{5} - \frac{17236984044954762166848161485882139600889915992498504417646227}{100375850547163313566098818489329089440692663904105484501361064} a^{4} + \frac{38956498637027609202843310821743511530692474649653467949784609}{100375850547163313566098818489329089440692663904105484501361064} a^{3} - \frac{15803727395088818432321036823098421406657132559923372365469451}{50187925273581656783049409244664544720346331952052742250680532} a^{2} - \frac{41582566951954560747106910524660206399536743628767474835690215}{100375850547163313566098818489329089440692663904105484501361064} a - \frac{5100363816852858012126646788561384742309976483954541155712131}{12546981318395414195762352311166136180086582988013185562670133}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{12}\times C_{24}$, which has order $2304$ (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:  \( 1293672.85735 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

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 $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.0.8268784250602433.5, 8.6.123414690307499.1, 8.2.1617217123.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$67$67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$