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Properties

Label 16.0.14906424550...3344.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{14}\cdot 11^{6}$
Root discriminant $43.24$
Ramified primes $2, 3, 11$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_2^3.(C_2\times D_4)$ (as 16T408)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![277, 596, -24, -2300, 2042, 4740, -1424, 20, 750, -724, 568, -324, 206, -68, 24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 24*x^14 - 68*x^13 + 206*x^12 - 324*x^11 + 568*x^10 - 724*x^9 + 750*x^8 + 20*x^7 - 1424*x^6 + 4740*x^5 + 2042*x^4 - 2300*x^3 - 24*x^2 + 596*x + 277)
 
gp: K = bnfinit(x^16 - 4*x^15 + 24*x^14 - 68*x^13 + 206*x^12 - 324*x^11 + 568*x^10 - 724*x^9 + 750*x^8 + 20*x^7 - 1424*x^6 + 4740*x^5 + 2042*x^4 - 2300*x^3 - 24*x^2 + 596*x + 277, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 24 x^{14} - 68 x^{13} + 206 x^{12} - 324 x^{11} + 568 x^{10} - 724 x^{9} + 750 x^{8} + 20 x^{7} - 1424 x^{6} + 4740 x^{5} + 2042 x^{4} - 2300 x^{3} - 24 x^{2} + 596 x + 277 \)

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:  \(149064245508482666116153344=2^{44}\cdot 3^{14}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.24$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{66} a^{14} - \frac{2}{33} a^{13} + \frac{1}{22} a^{12} + \frac{5}{66} a^{11} - \frac{2}{33} a^{8} + \frac{1}{33} a^{7} - \frac{13}{66} a^{6} + \frac{1}{3} a^{5} + \frac{5}{22} a^{4} - \frac{1}{66} a^{3} + \frac{10}{33} a + \frac{10}{33}$, $\frac{1}{165405444653198431489302} a^{15} - \frac{1600470107158099849}{9189191369622135082739} a^{14} + \frac{1873620393903115118863}{27567574108866405248217} a^{13} + \frac{3034531546777836802363}{165405444653198431489302} a^{12} - \frac{926214375538641280133}{18378382739244270165478} a^{11} - \frac{27928620943869985890}{835381033602012280249} a^{10} - \frac{11859946491361720052789}{165405444653198431489302} a^{9} - \frac{2691595099987138417511}{55135148217732810496434} a^{8} - \frac{3280860955349726064147}{18378382739244270165478} a^{7} + \frac{17214937420071135899410}{82702722326599215744651} a^{6} - \frac{3919225385371939083268}{27567574108866405248217} a^{5} - \frac{23838421140218728270235}{55135148217732810496434} a^{4} - \frac{5247628965816410198077}{165405444653198431489302} a^{3} + \frac{2736143082508932247367}{27567574108866405248217} a^{2} + \frac{2222655895789686982271}{18378382739244270165478} a - \frac{28845245998130905467283}{165405444653198431489302}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( 181988.985617 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.(C_2\times D_4)$ (as 16T408):

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 $C_2^3.(C_2\times D_4)$
Character table for $C_2^3.(C_2\times D_4)$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.4752.1, 4.0.13824.1, 4.0.50688.2, 8.8.763074183168.1, 8.0.190768545792.5, 8.0.23123460096.18

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
3Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$