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Properties

Label 16.0.85010002742...0144.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 3^{8}\cdot 13^{6}$
Root discriminant $15.24$
Ramified primes $2, 3, 13$
Class number $2$
Class group $[2]$
Galois group $C_2\wr C_2^2$ (as 16T128)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -324, 594, -702, 648, -522, 420, -336, 187, -58, 48, -64, 42, -22, 14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 14*x^14 - 22*x^13 + 42*x^12 - 64*x^11 + 48*x^10 - 58*x^9 + 187*x^8 - 336*x^7 + 420*x^6 - 522*x^5 + 648*x^4 - 702*x^3 + 594*x^2 - 324*x + 81)
 
gp: K = bnfinit(x^16 - 6*x^15 + 14*x^14 - 22*x^13 + 42*x^12 - 64*x^11 + 48*x^10 - 58*x^9 + 187*x^8 - 336*x^7 + 420*x^6 - 522*x^5 + 648*x^4 - 702*x^3 + 594*x^2 - 324*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 14 x^{14} - 22 x^{13} + 42 x^{12} - 64 x^{11} + 48 x^{10} - 58 x^{9} + 187 x^{8} - 336 x^{7} + 420 x^{6} - 522 x^{5} + 648 x^{4} - 702 x^{3} + 594 x^{2} - 324 x + 81 \)

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:  \(8501000274280710144=2^{28}\cdot 3^{8}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.24$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{4}{9} a^{7} - \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{3} a^{9} - \frac{4}{9} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} + \frac{1}{9} a^{5}$, $\frac{1}{27} a^{14} - \frac{1}{27} a^{12} - \frac{1}{27} a^{11} - \frac{1}{9} a^{10} - \frac{13}{27} a^{9} - \frac{4}{9} a^{8} + \frac{11}{27} a^{7} + \frac{1}{27} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{28067607} a^{15} + \frac{146675}{28067607} a^{14} + \frac{95153}{28067607} a^{13} + \frac{433082}{9355869} a^{12} + \frac{848743}{28067607} a^{11} - \frac{1997782}{28067607} a^{10} + \frac{1630078}{28067607} a^{9} - \frac{4393891}{28067607} a^{8} - \frac{8815027}{28067607} a^{7} - \frac{7563595}{28067607} a^{6} + \frac{1006508}{9355869} a^{5} + \frac{1381706}{3118623} a^{4} + \frac{64957}{3118623} a^{3} + \frac{564374}{3118623} a^{2} + \frac{44922}{1039541} a - \frac{446529}{1039541}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{4622585}{9355869} a^{15} - \frac{69926666}{28067607} a^{14} + \frac{41745143}{9355869} a^{13} - \frac{177588130}{28067607} a^{12} + \frac{402972554}{28067607} a^{11} - \frac{17989973}{1039541} a^{10} + \frac{159609947}{28067607} a^{9} - \frac{208335475}{9355869} a^{8} + \frac{1994823332}{28067607} a^{7} - \frac{2655441359}{28067607} a^{6} + \frac{1034285309}{9355869} a^{5} - \frac{1362972964}{9355869} a^{4} + \frac{534069305}{3118623} a^{3} - \frac{533159063}{3118623} a^{2} + \frac{122141796}{1039541} a - \frac{37734415}{1039541} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2343.64342971 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T128):

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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.4.7488.1, 4.0.7488.1, \(\Q(\zeta_{12})\), 8.0.4313088.1, 8.0.56070144.2, 8.0.728911872.3

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$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}$