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Properties

Label 16.0.10737418240...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{10}$
Root discriminant $15.47$
Ramified primes $2, 5$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_4\wr C_2$ (as 16T111)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 68, -252, 666, -1248, 1748, -1964, 1804, -1364, 884, -460, 214, -72, 24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 24*x^14 - 72*x^13 + 214*x^12 - 460*x^11 + 884*x^10 - 1364*x^9 + 1804*x^8 - 1964*x^7 + 1748*x^6 - 1248*x^5 + 666*x^4 - 252*x^3 + 68*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 24*x^14 - 72*x^13 + 214*x^12 - 460*x^11 + 884*x^10 - 1364*x^9 + 1804*x^8 - 1964*x^7 + 1748*x^6 - 1248*x^5 + 666*x^4 - 252*x^3 + 68*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 24 x^{14} - 72 x^{13} + 214 x^{12} - 460 x^{11} + 884 x^{10} - 1364 x^{9} + 1804 x^{8} - 1964 x^{7} + 1748 x^{6} - 1248 x^{5} + 666 x^{4} - 252 x^{3} + 68 x^{2} - 12 x + 1 \)

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:  \(10737418240000000000=2^{40}\cdot 5^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.47$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{95} a^{14} - \frac{4}{19} a^{13} - \frac{3}{19} a^{12} + \frac{33}{95} a^{11} + \frac{36}{95} a^{10} + \frac{37}{95} a^{9} + \frac{3}{95} a^{8} + \frac{6}{19} a^{7} - \frac{2}{5} a^{6} + \frac{34}{95} a^{5} + \frac{26}{95} a^{4} - \frac{23}{95} a^{2} + \frac{21}{95} a + \frac{9}{95}$, $\frac{1}{18780835} a^{15} + \frac{13151}{18780835} a^{14} - \frac{89271}{197693} a^{13} - \frac{1098322}{18780835} a^{12} - \frac{6952046}{18780835} a^{11} - \frac{8105352}{18780835} a^{10} - \frac{946413}{3756167} a^{9} - \frac{1930757}{18780835} a^{8} - \frac{7219918}{18780835} a^{7} - \frac{1363729}{18780835} a^{6} - \frac{79931}{3756167} a^{5} - \frac{9109009}{18780835} a^{4} + \frac{6948467}{18780835} a^{3} + \frac{759188}{18780835} a^{2} + \frac{91724}{3756167} a + \frac{5881904}{18780835}$

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:  \( \frac{2113552}{988465} a^{15} - \frac{153632183}{18780835} a^{14} + \frac{187257492}{3756167} a^{13} - \frac{2724788926}{18780835} a^{12} + \frac{8099503024}{18780835} a^{11} - \frac{16997453642}{18780835} a^{10} + \frac{32348630773}{18780835} a^{9} - \frac{48712360419}{18780835} a^{8} + \frac{63133096126}{18780835} a^{7} - \frac{3503921706}{988465} a^{6} + \frac{56902307956}{18780835} a^{5} - \frac{38391657518}{18780835} a^{4} + \frac{976183614}{988465} a^{3} - \frac{5915433163}{18780835} a^{2} + \frac{1307987479}{18780835} a - \frac{161734922}{18780835} \) (order $8$)
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:  \( 2596.24415474 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4\wr C_2$ (as 16T111):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $C_2\times C_4\wr C_2$
Character table for $C_2\times C_4\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.320.1, 4.0.1280.1, \(\Q(\zeta_{8})\), 8.0.204800000.1, 8.0.819200000.1, 8.0.6553600.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$