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Properties

Label 16.0.93210154223...6117.2
Degree $16$
Signature $[0, 8]$
Discriminant $7^{10}\cdot 53^{9}$
Root discriminant $31.48$
Ramified primes $7, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12604, -28339, 47262, -45841, 32970, -15859, 5212, -699, -245, 214, -55, -25, 56, -38, 19, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 19*x^14 - 38*x^13 + 56*x^12 - 25*x^11 - 55*x^10 + 214*x^9 - 245*x^8 - 699*x^7 + 5212*x^6 - 15859*x^5 + 32970*x^4 - 45841*x^3 + 47262*x^2 - 28339*x + 12604)
 
gp: K = bnfinit(x^16 - 5*x^15 + 19*x^14 - 38*x^13 + 56*x^12 - 25*x^11 - 55*x^10 + 214*x^9 - 245*x^8 - 699*x^7 + 5212*x^6 - 15859*x^5 + 32970*x^4 - 45841*x^3 + 47262*x^2 - 28339*x + 12604, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 19 x^{14} - 38 x^{13} + 56 x^{12} - 25 x^{11} - 55 x^{10} + 214 x^{9} - 245 x^{8} - 699 x^{7} + 5212 x^{6} - 15859 x^{5} + 32970 x^{4} - 45841 x^{3} + 47262 x^{2} - 28339 x + 12604 \)

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:  \(932101542235441877906117=7^{10}\cdot 53^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.48$
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:  $7, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{32} a^{14} + \frac{1}{32} a^{13} + \frac{1}{32} a^{11} - \frac{1}{16} a^{10} + \frac{1}{16} a^{9} + \frac{7}{32} a^{8} + \frac{7}{16} a^{7} - \frac{1}{2} a^{6} + \frac{7}{32} a^{5} - \frac{5}{16} a^{4} + \frac{1}{16} a^{3} - \frac{1}{2} a^{2} - \frac{3}{32} a - \frac{1}{8}$, $\frac{1}{281951183128224663203072} a^{15} - \frac{680056942127358315271}{140975591564112331601536} a^{14} + \frac{13996678549280553004401}{281951183128224663203072} a^{13} - \frac{14034677009644858014719}{281951183128224663203072} a^{12} - \frac{34403462733319561771153}{281951183128224663203072} a^{11} - \frac{1905136777464448258063}{8810974472757020725096} a^{10} + \frac{1304990283006568686249}{281951183128224663203072} a^{9} + \frac{30897687494140352063845}{281951183128224663203072} a^{8} + \frac{44228282098997455525839}{140975591564112331601536} a^{7} + \frac{102993850290778137519095}{281951183128224663203072} a^{6} + \frac{65619656576875647045485}{281951183128224663203072} a^{5} - \frac{81894942748535726737}{2711069068540621761568} a^{4} + \frac{54193458560046649048505}{140975591564112331601536} a^{3} + \frac{57128843575762111062509}{281951183128224663203072} a^{2} - \frac{90017468275715073027767}{281951183128224663203072} a + \frac{33691461970905609209207}{70487795782056165800768}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

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:  \( -1 \) (order $2$)
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3639322.45277 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.2597.2, 8.0.357453677.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ $16$ $16$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$53$53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.4.3.3$x^{4} + 106$$4$$1$$3$$C_4$$[\ ]_{4}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$