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Properties

Label 16.0.62459261715...2809.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 173^{6}$
Root discriminant $47.29$
Ramified primes $13, 173$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, 6561, 18549, 18603, 13689, 3882, 5812, -13482, 9784, -3520, 954, -291, 35, 6, 12, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 12*x^14 + 6*x^13 + 35*x^12 - 291*x^11 + 954*x^10 - 3520*x^9 + 9784*x^8 - 13482*x^7 + 5812*x^6 + 3882*x^5 + 13689*x^4 + 18603*x^3 + 18549*x^2 + 6561*x + 729)
 
gp: K = bnfinit(x^16 - 7*x^15 + 12*x^14 + 6*x^13 + 35*x^12 - 291*x^11 + 954*x^10 - 3520*x^9 + 9784*x^8 - 13482*x^7 + 5812*x^6 + 3882*x^5 + 13689*x^4 + 18603*x^3 + 18549*x^2 + 6561*x + 729, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 12 x^{14} + 6 x^{13} + 35 x^{12} - 291 x^{11} + 954 x^{10} - 3520 x^{9} + 9784 x^{8} - 13482 x^{7} + 5812 x^{6} + 3882 x^{5} + 13689 x^{4} + 18603 x^{3} + 18549 x^{2} + 6561 x + 729 \)

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:  \(624592617158605666432592809=13^{12}\cdot 173^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.29$
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:  $13, 173$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{1}{9} a^{4} - \frac{4}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{2}{9} a^{4} + \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{783} a^{13} - \frac{43}{783} a^{12} + \frac{10}{261} a^{11} - \frac{7}{261} a^{10} - \frac{4}{783} a^{9} - \frac{14}{261} a^{8} - \frac{11}{87} a^{7} - \frac{40}{783} a^{6} + \frac{382}{783} a^{5} + \frac{19}{261} a^{4} - \frac{137}{783} a^{3} + \frac{98}{261} a^{2} + \frac{10}{29} a + \frac{3}{29}$, $\frac{1}{7047} a^{14} - \frac{4}{7047} a^{13} + \frac{20}{783} a^{12} + \frac{122}{2349} a^{11} + \frac{134}{7047} a^{10} - \frac{8}{2349} a^{9} + \frac{13}{261} a^{8} + \frac{449}{7047} a^{7} - \frac{830}{7047} a^{6} - \frac{119}{2349} a^{5} + \frac{520}{7047} a^{4} + \frac{347}{2349} a^{3} - \frac{233}{783} a^{2} - \frac{71}{261} a + \frac{10}{87}$, $\frac{1}{44841330903505555048527} a^{15} - \frac{314094313026832339}{44841330903505555048527} a^{14} - \frac{4991391267700520959}{14947110301168518349509} a^{13} + \frac{9635463551724837494}{14947110301168518349509} a^{12} - \frac{867473325038393132428}{44841330903505555048527} a^{11} - \frac{109456175084715506}{57268621843557541569} a^{10} + \frac{223475185093276231900}{4982370100389506116503} a^{9} - \frac{2019875974241981654557}{44841330903505555048527} a^{8} - \frac{407918327767086627401}{44841330903505555048527} a^{7} + \frac{1194415151050242234923}{14947110301168518349509} a^{6} - \frac{18252577353169550346149}{44841330903505555048527} a^{5} - \frac{2126607845502549659651}{4982370100389506116503} a^{4} - \frac{711792055048239074362}{1660790033463168705501} a^{3} - \frac{111304154892739316410}{1660790033463168705501} a^{2} - \frac{3897179756067450149}{19089540614519180523} a - \frac{1854828442857567488}{184532225940352078389}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 7203842.04142 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

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 $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.29237.1, 4.0.2197.1, 4.0.380081.1, 8.0.144461566561.1

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 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed
$173$173.4.2.2$x^{4} - 173 x^{2} + 149645$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
173.4.0.1$x^{4} - x + 26$$1$$4$$0$$C_4$$[\ ]^{4}$
173.8.4.1$x^{8} + 1556308 x^{4} - 5177717 x^{2} + 605523647716$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$