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Properties

Label 16.0.23043642001...0976.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{72}\cdot 47^{4}$
Root discriminant $59.25$
Ramified primes $2, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![314044, 596864, 610432, 347264, 203552, 63200, 35808, 2240, 2860, -736, 640, -96, 8, -48, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^13 + 8*x^12 - 96*x^11 + 640*x^10 - 736*x^9 + 2860*x^8 + 2240*x^7 + 35808*x^6 + 63200*x^5 + 203552*x^4 + 347264*x^3 + 610432*x^2 + 596864*x + 314044)
 
gp: K = bnfinit(x^16 - 48*x^13 + 8*x^12 - 96*x^11 + 640*x^10 - 736*x^9 + 2860*x^8 + 2240*x^7 + 35808*x^6 + 63200*x^5 + 203552*x^4 + 347264*x^3 + 610432*x^2 + 596864*x + 314044, 1)
 

Normalized defining polynomial

\( x^{16} - 48 x^{13} + 8 x^{12} - 96 x^{11} + 640 x^{10} - 736 x^{9} + 2860 x^{8} + 2240 x^{7} + 35808 x^{6} + 63200 x^{5} + 203552 x^{4} + 347264 x^{3} + 610432 x^{2} + 596864 x + 314044 \)

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:  \(23043642001495833226013310976=2^{72}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.25$
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:  $2, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{68} a^{14} + \frac{3}{68} a^{13} + \frac{1}{68} a^{12} - \frac{1}{68} a^{11} - \frac{3}{68} a^{10} + \frac{5}{68} a^{9} + \frac{4}{17} a^{8} + \frac{5}{17} a^{7} + \frac{1}{17} a^{6} - \frac{4}{17} a^{5} - \frac{3}{34} a^{4} + \frac{1}{34} a^{3} + \frac{7}{34} a^{2} + \frac{7}{34} a + \frac{7}{17}$, $\frac{1}{1121241444000832597319845895217988} a^{15} + \frac{4226640409055386679657754462403}{1121241444000832597319845895217988} a^{14} + \frac{67044415738198130482890833818223}{560620722000416298659922947608994} a^{13} + \frac{27652033839188615777165827810803}{560620722000416298659922947608994} a^{12} - \frac{9077782244964806588779102380353}{1121241444000832597319845895217988} a^{11} - \frac{12614557193630522532485714219300}{280310361000208149329961473804497} a^{10} + \frac{506213287588830262377284148735}{16488844764718126431174204341441} a^{9} + \frac{63403420942223845460566614138027}{560620722000416298659922947608994} a^{8} - \frac{60842895234430227927714176380739}{280310361000208149329961473804497} a^{7} - \frac{232892762154988691553114681682357}{560620722000416298659922947608994} a^{6} - \frac{15637058560463393148225065225766}{280310361000208149329961473804497} a^{5} - \frac{95786710198904730119966571842742}{280310361000208149329961473804497} a^{4} - \frac{193690395334927125652624350345779}{560620722000416298659922947608994} a^{3} + \frac{46659130643695276535677649076311}{280310361000208149329961473804497} a^{2} - \frac{302382745986089982624711870602}{763788449591847818337769683391} a - \frac{6425376263132969020701607528986}{280310361000208149329961473804497}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 43800097.3343 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.0.2147483648.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
47Data not computed