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Properties

Label 16.0.28147497671...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{52}\cdot 5^{4}$
Root discriminant $14.23$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $D_4^2.C_2$ (as 16T376)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -16, 48, -64, 28, -8, 144, -448, 758, -872, 752, -504, 268, -112, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 268*x^12 - 504*x^11 + 752*x^10 - 872*x^9 + 758*x^8 - 448*x^7 + 144*x^6 - 8*x^5 + 28*x^4 - 64*x^3 + 48*x^2 - 16*x + 2)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 268*x^12 - 504*x^11 + 752*x^10 - 872*x^9 + 758*x^8 - 448*x^7 + 144*x^6 - 8*x^5 + 28*x^4 - 64*x^3 + 48*x^2 - 16*x + 2, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 112 x^{13} + 268 x^{12} - 504 x^{11} + 752 x^{10} - 872 x^{9} + 758 x^{8} - 448 x^{7} + 144 x^{6} - 8 x^{5} + 28 x^{4} - 64 x^{3} + 48 x^{2} - 16 x + 2 \)

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:  \(2814749767106560000=2^{52}\cdot 5^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{52194077} a^{15} + \frac{11323419}{52194077} a^{14} - \frac{8927174}{52194077} a^{13} - \frac{2998661}{52194077} a^{12} - \frac{1168244}{52194077} a^{11} + \frac{10968881}{52194077} a^{10} + \frac{17732425}{52194077} a^{9} + \frac{5336832}{52194077} a^{8} - \frac{14279964}{52194077} a^{7} + \frac{7928233}{52194077} a^{6} + \frac{15681249}{52194077} a^{5} - \frac{24291533}{52194077} a^{4} + \frac{11475361}{52194077} a^{3} + \frac{21857809}{52194077} a^{2} - \frac{229972}{52194077} a - \frac{4264376}{52194077}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{30777940}{4014929} a^{15} - \frac{233771838}{4014929} a^{14} + \frac{1014044516}{4014929} a^{13} - \frac{3041386940}{4014929} a^{12} + \frac{7036749796}{4014929} a^{11} - \frac{12718799672}{4014929} a^{10} + \frac{18117376204}{4014929} a^{9} - \frac{19709715159}{4014929} a^{8} + \frac{15620712824}{4014929} a^{7} - \frac{7726528020}{4014929} a^{6} + \frac{1476085012}{4014929} a^{5} + \frac{296078654}{4014929} a^{4} + \frac{979674728}{4014929} a^{3} - \frac{1579394656}{4014929} a^{2} + \frac{865678528}{4014929} a - \frac{161729599}{4014929} \) (order $4$)
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:  \( 1678.98591432 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4^2.C_2$ (as 16T376):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $D_4^2.C_2$
Character table for $D_4^2.C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.512.1, 4.0.1280.1, 4.0.2560.2, 8.0.83886080.2, 8.0.83886080.1, 8.0.104857600.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$