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Properties

Label 16.0.35007428516...6169.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{6}\cdot 29^{14}$
Root discriminant $39.49$
Ramified primes $7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![343, -686, -2646, 6713, 14168, 1204, 13249, 2997, 1652, 2441, -262, 310, 88, -34, 23, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 23*x^14 - 34*x^13 + 88*x^12 + 310*x^11 - 262*x^10 + 2441*x^9 + 1652*x^8 + 2997*x^7 + 13249*x^6 + 1204*x^5 + 14168*x^4 + 6713*x^3 - 2646*x^2 - 686*x + 343)
 
gp: K = bnfinit(x^16 - 4*x^15 + 23*x^14 - 34*x^13 + 88*x^12 + 310*x^11 - 262*x^10 + 2441*x^9 + 1652*x^8 + 2997*x^7 + 13249*x^6 + 1204*x^5 + 14168*x^4 + 6713*x^3 - 2646*x^2 - 686*x + 343, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 23 x^{14} - 34 x^{13} + 88 x^{12} + 310 x^{11} - 262 x^{10} + 2441 x^{9} + 1652 x^{8} + 2997 x^{7} + 13249 x^{6} + 1204 x^{5} + 14168 x^{4} + 6713 x^{3} - 2646 x^{2} - 686 x + 343 \)

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:  \(35007428516075131079076169=7^{6}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.49$
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:  $7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{1}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2}$, $\frac{1}{35} a^{12} - \frac{2}{35} a^{11} - \frac{2}{7} a^{10} + \frac{8}{35} a^{9} - \frac{8}{35} a^{8} - \frac{3}{7} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{13}{35} a^{4} + \frac{17}{35} a^{3} + \frac{17}{35} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{35} a^{13} + \frac{1}{35} a^{11} + \frac{3}{35} a^{10} + \frac{8}{35} a^{9} - \frac{16}{35} a^{8} + \frac{1}{5} a^{7} + \frac{1}{7} a^{6} - \frac{17}{35} a^{4} - \frac{4}{35} a^{3} + \frac{12}{35} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{245} a^{14} + \frac{3}{245} a^{13} + \frac{2}{245} a^{12} - \frac{6}{245} a^{11} + \frac{102}{245} a^{10} + \frac{51}{245} a^{9} + \frac{116}{245} a^{8} - \frac{79}{245} a^{7} + \frac{16}{35} a^{6} - \frac{111}{245} a^{5} + \frac{103}{245} a^{4} + \frac{16}{35} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5766147926254449303458245} a^{15} - \frac{8575921835487406051004}{5766147926254449303458245} a^{14} + \frac{21857365340270785781839}{5766147926254449303458245} a^{13} + \frac{76884390460337862576261}{5766147926254449303458245} a^{12} + \frac{45595874301574922340849}{1153229585250889860691649} a^{11} - \frac{1438804602103904502317614}{5766147926254449303458245} a^{10} - \frac{2458145218574740126564014}{5766147926254449303458245} a^{9} - \frac{135497555184320526036572}{5766147926254449303458245} a^{8} + \frac{637689856664563260158676}{5766147926254449303458245} a^{7} + \frac{209513996233466814531167}{5766147926254449303458245} a^{6} + \frac{913944276051441443887721}{5766147926254449303458245} a^{5} + \frac{2605467362810405238680463}{5766147926254449303458245} a^{4} - \frac{170094750468729645175829}{823735418036349900494035} a^{3} + \frac{322821814731855295326831}{823735418036349900494035} a^{2} - \frac{50158316579557465423029}{117676488290907128642005} a + \frac{7748786441808794517084}{117676488290907128642005}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1690145.93466 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.24389.1, 4.2.5887.1, 4.2.170723.1, 8.0.845243939141.1 x2, 8.0.29146342729.1

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 8 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29Data not computed