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Properties

Label 16.0.26873856000...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}$
Root discriminant $16.38$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -14, 98, -506, 1533, -1654, 940, -100, -274, 320, -190, 44, 28, -34, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 34*x^13 + 28*x^12 + 44*x^11 - 190*x^10 + 320*x^9 - 274*x^8 - 100*x^7 + 940*x^6 - 1654*x^5 + 1533*x^4 - 506*x^3 + 98*x^2 - 14*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 34*x^13 + 28*x^12 + 44*x^11 - 190*x^10 + 320*x^9 - 274*x^8 - 100*x^7 + 940*x^6 - 1654*x^5 + 1533*x^4 - 506*x^3 + 98*x^2 - 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 34 x^{13} + 28 x^{12} + 44 x^{11} - 190 x^{10} + 320 x^{9} - 274 x^{8} - 100 x^{7} + 940 x^{6} - 1654 x^{5} + 1533 x^{4} - 506 x^{3} + 98 x^{2} - 14 x + 1 \)

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:  \(26873856000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.38$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} - \frac{2}{5}$, $\frac{1}{2705} a^{14} - \frac{269}{2705} a^{13} + \frac{204}{2705} a^{12} - \frac{203}{2705} a^{11} - \frac{199}{2705} a^{10} - \frac{811}{2705} a^{9} - \frac{608}{2705} a^{8} + \frac{784}{2705} a^{7} - \frac{1098}{2705} a^{6} - \frac{1171}{2705} a^{5} - \frac{89}{2705} a^{4} - \frac{427}{2705} a^{3} - \frac{857}{2705} a^{2} - \frac{1078}{2705} a + \frac{1171}{2705}$, $\frac{1}{25180470641525} a^{15} - \frac{1626803014}{25180470641525} a^{14} - \frac{469248576576}{5036094128305} a^{13} - \frac{61346708769}{614157820525} a^{12} + \frac{1832919615989}{5036094128305} a^{11} + \frac{466863840539}{25180470641525} a^{10} - \frac{1185745047412}{25180470641525} a^{9} - \frac{9636407693724}{25180470641525} a^{8} - \frac{8810680141152}{25180470641525} a^{7} - \frac{76873618189}{614157820525} a^{6} - \frac{11860860779713}{25180470641525} a^{5} - \frac{732948505078}{5036094128305} a^{4} - \frac{6062774949307}{25180470641525} a^{3} + \frac{1133026641509}{5036094128305} a^{2} + \frac{12445176171283}{25180470641525} a + \frac{6313761253727}{25180470641525}$

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:  \( -\frac{11640438763398}{25180470641525} a^{15} + \frac{68969248263277}{25180470641525} a^{14} - \frac{40852475417462}{5036094128305} a^{13} + \frac{9270077165087}{614157820525} a^{12} - \frac{59300863001964}{5036094128305} a^{11} - \frac{536005878035192}{25180470641525} a^{10} + \frac{2171950122055041}{25180470641525} a^{9} - \frac{3558803824105198}{25180470641525} a^{8} + \frac{2913399480462686}{25180470641525} a^{7} + \frac{34088351828247}{614157820525} a^{6} - \frac{10847915221131086}{25180470641525} a^{5} + \frac{3685971985291163}{5036094128305} a^{4} - \frac{16407668437748949}{25180470641525} a^{3} + \frac{919361748012508}{5036094128305} a^{2} - \frac{717106289866809}{25180470641525} a + \frac{81132845508049}{25180470641525} \) (order $20$)
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:  \( 9740.05338886 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 4.2.3600.1 x2, 4.0.2880.1 x2, 4.0.72000.1 x2, 4.2.18000.1 x2, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 8.0.207360000.5, 8.0.5184000000.15, \(\Q(\zeta_{20})\), 8.0.324000000.5 x2, 8.4.5184000000.2 x2

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

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$