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Properties

Label 16.0.12478436359...5625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 59^{10}$
Root discriminant $42.76$
Ramified primes $5, 59$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T330)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![652231, -743902, 557706, -351523, 204035, -79152, 20768, -5889, 3693, -2631, 1128, -446, 179, -74, 27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 27*x^14 - 74*x^13 + 179*x^12 - 446*x^11 + 1128*x^10 - 2631*x^9 + 3693*x^8 - 5889*x^7 + 20768*x^6 - 79152*x^5 + 204035*x^4 - 351523*x^3 + 557706*x^2 - 743902*x + 652231)
 
gp: K = bnfinit(x^16 - 6*x^15 + 27*x^14 - 74*x^13 + 179*x^12 - 446*x^11 + 1128*x^10 - 2631*x^9 + 3693*x^8 - 5889*x^7 + 20768*x^6 - 79152*x^5 + 204035*x^4 - 351523*x^3 + 557706*x^2 - 743902*x + 652231, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 27 x^{14} - 74 x^{13} + 179 x^{12} - 446 x^{11} + 1128 x^{10} - 2631 x^{9} + 3693 x^{8} - 5889 x^{7} + 20768 x^{6} - 79152 x^{5} + 204035 x^{4} - 351523 x^{3} + 557706 x^{2} - 743902 x + 652231 \)

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:  \(124784363598789404541015625=5^{12}\cdot 59^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.76$
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:  $5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} + \frac{1}{15} a^{11} + \frac{1}{3} a^{10} + \frac{1}{15} a^{8} - \frac{1}{3} a^{7} + \frac{4}{15} a^{6} - \frac{1}{3} a^{5} - \frac{4}{15} a^{4} - \frac{1}{3} a^{2} + \frac{2}{5} a - \frac{4}{15}$, $\frac{1}{15} a^{13} + \frac{4}{15} a^{11} - \frac{1}{3} a^{10} + \frac{1}{15} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{15} a^{5} + \frac{4}{15} a^{4} - \frac{1}{3} a^{3} - \frac{4}{15} a^{2} + \frac{1}{3} a + \frac{4}{15}$, $\frac{1}{105} a^{14} + \frac{2}{105} a^{12} + \frac{8}{105} a^{11} + \frac{17}{35} a^{10} + \frac{8}{35} a^{9} - \frac{38}{105} a^{8} - \frac{29}{105} a^{7} + \frac{38}{105} a^{6} - \frac{1}{105} a^{5} - \frac{4}{35} a^{4} + \frac{41}{105} a^{3} - \frac{1}{7} a^{2} + \frac{22}{105} a - \frac{37}{105}$, $\frac{1}{243408506351774261896294356399212595} a^{15} - \frac{133557696558451216142754304291249}{81136168783924753965431452133070865} a^{14} + \frac{355226804228687346781212880041184}{48681701270354852379258871279842519} a^{13} - \frac{891049943677822108837977736983197}{243408506351774261896294356399212595} a^{12} - \frac{69836420048923147904254733686289554}{243408506351774261896294356399212595} a^{11} - \frac{41195410417791072787554788515939658}{243408506351774261896294356399212595} a^{10} - \frac{7244052575544693385009229293202318}{243408506351774261896294356399212595} a^{9} - \frac{18621226227733474981280313841122479}{81136168783924753965431452133070865} a^{8} - \frac{9879046465689393082598245796079752}{243408506351774261896294356399212595} a^{7} - \frac{57032353838397839171164992389492183}{243408506351774261896294356399212595} a^{6} + \frac{14045592478042892070794060593855493}{243408506351774261896294356399212595} a^{5} + \frac{29492098892689370219092631329094802}{81136168783924753965431452133070865} a^{4} + \frac{1326124724179376786469375540236563}{11590881254846393423633064590438695} a^{3} + \frac{1648349445409123350943005918872857}{16227233756784950793086290426614173} a^{2} + \frac{76296733267795216594293225558158738}{243408506351774261896294356399212595} a + \frac{24225403077207726034503376420931198}{48681701270354852379258871279842519}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 790297.748928 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T330):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1475.1, 8.0.37866753125.1, 8.2.3209046875.2, 8.2.2234138434375.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
59Data not computed