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Properties

Label 16.0.89791815397...896.10
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 13^{8}$
Root discriminant $17.66$
Ramified primes $2, 3, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\wr C_2^2$ (as 16T149)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2401, -6664, 9248, -8416, 5784, -3410, 1936, -1238, 890, -584, 290, -110, 49, -32, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 32*x^13 + 49*x^12 - 110*x^11 + 290*x^10 - 584*x^9 + 890*x^8 - 1238*x^7 + 1936*x^6 - 3410*x^5 + 5784*x^4 - 8416*x^3 + 9248*x^2 - 6664*x + 2401)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 32*x^13 + 49*x^12 - 110*x^11 + 290*x^10 - 584*x^9 + 890*x^8 - 1238*x^7 + 1936*x^6 - 3410*x^5 + 5784*x^4 - 8416*x^3 + 9248*x^2 - 6664*x + 2401, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 32 x^{13} + 49 x^{12} - 110 x^{11} + 290 x^{10} - 584 x^{9} + 890 x^{8} - 1238 x^{7} + 1936 x^{6} - 3410 x^{5} + 5784 x^{4} - 8416 x^{3} + 9248 x^{2} - 6664 x + 2401 \)

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:  \(89791815397090000896=2^{24}\cdot 3^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.66$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{559} a^{14} + \frac{158}{559} a^{13} + \frac{184}{559} a^{12} - \frac{67}{559} a^{11} - \frac{57}{559} a^{10} - \frac{47}{559} a^{9} - \frac{4}{559} a^{8} + \frac{264}{559} a^{7} + \frac{153}{559} a^{6} - \frac{246}{559} a^{5} - \frac{261}{559} a^{4} + \frac{229}{559} a^{3} + \frac{264}{559} a^{2} + \frac{161}{559} a - \frac{187}{559}$, $\frac{1}{2349147166203929} a^{15} - \frac{683457291222}{2349147166203929} a^{14} - \frac{83049688233431}{2349147166203929} a^{13} + \frac{208940879455983}{2349147166203929} a^{12} - \frac{12915363398481}{47941778902121} a^{11} - \frac{735884117363659}{2349147166203929} a^{10} - \frac{147134164215735}{2349147166203929} a^{9} - \frac{773461351938481}{2349147166203929} a^{8} + \frac{1010619624400814}{2349147166203929} a^{7} + \frac{197310952505374}{2349147166203929} a^{6} + \frac{503118258335998}{2349147166203929} a^{5} - \frac{433446557818338}{2349147166203929} a^{4} - \frac{567699269410278}{2349147166203929} a^{3} + \frac{492438966559027}{2349147166203929} a^{2} + \frac{26519982593966}{2349147166203929} a + \frac{22362127586711}{47941778902121}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( \frac{581198026212}{180703628169533} a^{15} - \frac{13476255212736}{180703628169533} a^{14} + \frac{47212645170834}{180703628169533} a^{13} - \frac{85188239392301}{180703628169533} a^{12} + \frac{1799005467597}{3687829146317} a^{11} - \frac{217068316596175}{180703628169533} a^{10} + \frac{720636009602982}{180703628169533} a^{9} - \frac{1498169100220825}{180703628169533} a^{8} + \frac{2063405717241483}{180703628169533} a^{7} - \frac{2596147975356398}{180703628169533} a^{6} + \frac{4392578391491888}{180703628169533} a^{5} - \frac{7857060417195051}{180703628169533} a^{4} + \frac{322483735291443}{4202409957431} a^{3} - \frac{20598297783832655}{180703628169533} a^{2} + \frac{21367928742345961}{180703628169533} a - \frac{228988953256037}{3687829146317} \) (order $12$)
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:  \( 6351.23209965 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T149):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.0.117.1, \(\Q(\zeta_{12})\), 4.0.1872.1, 8.0.592240896.3 x2, 8.0.728911872.3 x2, 8.0.3504384.2

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$