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Properties

Label 16.0.18467573221...4016.9
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}$
Root discriminant $13.86$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
Galois group $C_2^2:D_4$ (as 16T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34, -104, 128, -16, -184, 312, -264, 56, 190, -328, 316, -232, 148, -80, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 148*x^12 - 232*x^11 + 316*x^10 - 328*x^9 + 190*x^8 + 56*x^7 - 264*x^6 + 312*x^5 - 184*x^4 - 16*x^3 + 128*x^2 - 104*x + 34)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 80*x^13 + 148*x^12 - 232*x^11 + 316*x^10 - 328*x^9 + 190*x^8 + 56*x^7 - 264*x^6 + 312*x^5 - 184*x^4 - 16*x^3 + 128*x^2 - 104*x + 34, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 80 x^{13} + 148 x^{12} - 232 x^{11} + 316 x^{10} - 328 x^{9} + 190 x^{8} + 56 x^{7} - 264 x^{6} + 312 x^{5} - 184 x^{4} - 16 x^{3} + 128 x^{2} - 104 x + 34 \)

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:  \(1846757322198614016=2^{48}\cdot 3^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.86$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{663} a^{14} - \frac{217}{663} a^{13} + \frac{37}{221} a^{12} + \frac{50}{663} a^{11} - \frac{77}{221} a^{10} + \frac{31}{221} a^{9} + \frac{14}{39} a^{8} - \frac{38}{221} a^{7} + \frac{4}{221} a^{6} - \frac{19}{663} a^{5} + \frac{101}{663} a^{4} + \frac{1}{13} a^{3} - \frac{66}{221} a^{2} - \frac{148}{663} a - \frac{16}{39}$, $\frac{1}{2264039583} a^{15} + \frac{287533}{754679861} a^{14} + \frac{905259113}{2264039583} a^{13} - \frac{283499839}{2264039583} a^{12} + \frac{1003090976}{2264039583} a^{11} - \frac{376841909}{754679861} a^{10} + \frac{628807663}{2264039583} a^{9} + \frac{800015347}{2264039583} a^{8} + \frac{254862397}{754679861} a^{7} + \frac{636030188}{2264039583} a^{6} - \frac{54766865}{174156891} a^{5} + \frac{94856483}{2264039583} a^{4} + \frac{1204224}{20396753} a^{3} + \frac{864042299}{2264039583} a^{2} + \frac{233954382}{754679861} a - \frac{24534424}{133178799}$

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:  \( \frac{11191346}{25438647} a^{15} - \frac{78603037}{25438647} a^{14} + \frac{280207268}{25438647} a^{13} - \frac{613395410}{25438647} a^{12} + \frac{1029215738}{25438647} a^{11} - \frac{511433269}{8479549} a^{10} + \frac{150175142}{1956819} a^{9} - \frac{1638437435}{25438647} a^{8} + \frac{125476722}{8479549} a^{7} + \frac{1085654608}{25438647} a^{6} - \frac{614354152}{8479549} a^{5} + \frac{1541564555}{25438647} a^{4} - \frac{3475180}{229177} a^{3} - \frac{671265728}{25438647} a^{2} + \frac{778370794}{25438647} a - \frac{19288885}{1496391} \) (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:  \( 1142.12354201 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.4608.1 x2, 4.2.9216.1 x2, 4.0.3072.1, \(\Q(\zeta_{8})\), 4.0.3072.2, 8.0.37748736.1, 8.0.9437184.1, 8.0.339738624.8

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$