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Properties

Label 16.16.1929403689...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $3^{12}\cdot 5^{14}\cdot 29^{6}$
Root discriminant $32.95$
Ramified primes $3, 5, 29$
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![1, -42, 614, -3867, 11363, -15687, 6236, 8349, -9797, 1383, 2569, -1116, -142, 156, -14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 14*x^14 + 156*x^13 - 142*x^12 - 1116*x^11 + 2569*x^10 + 1383*x^9 - 9797*x^8 + 8349*x^7 + 6236*x^6 - 15687*x^5 + 11363*x^4 - 3867*x^3 + 614*x^2 - 42*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 - 14*x^14 + 156*x^13 - 142*x^12 - 1116*x^11 + 2569*x^10 + 1383*x^9 - 9797*x^8 + 8349*x^7 + 6236*x^6 - 15687*x^5 + 11363*x^4 - 3867*x^3 + 614*x^2 - 42*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 14 x^{14} + 156 x^{13} - 142 x^{12} - 1116 x^{11} + 2569 x^{10} + 1383 x^{9} - 9797 x^{8} + 8349 x^{7} + 6236 x^{6} - 15687 x^{5} + 11363 x^{4} - 3867 x^{3} + 614 x^{2} - 42 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1929403689792242431640625=3^{12}\cdot 5^{14}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.95$
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:  $3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1639} a^{14} + \frac{486}{1639} a^{13} - \frac{452}{1639} a^{12} - \frac{815}{1639} a^{11} - \frac{224}{1639} a^{10} + \frac{615}{1639} a^{9} + \frac{274}{1639} a^{8} + \frac{58}{1639} a^{7} - \frac{4}{11} a^{6} + \frac{206}{1639} a^{5} - \frac{437}{1639} a^{4} + \frac{120}{1639} a^{3} + \frac{346}{1639} a^{2} - \frac{392}{1639} a - \frac{557}{1639}$, $\frac{1}{25861781} a^{15} - \frac{1}{75841} a^{14} - \frac{343370}{25861781} a^{13} - \frac{3912996}{25861781} a^{12} - \frac{8160429}{25861781} a^{11} + \frac{3265544}{25861781} a^{10} - \frac{8706609}{25861781} a^{9} - \frac{2604729}{25861781} a^{8} - \frac{2797165}{25861781} a^{7} + \frac{5356011}{25861781} a^{6} + \frac{4311866}{25861781} a^{5} - \frac{12909464}{25861781} a^{4} - \frac{1162605}{25861781} a^{3} - \frac{1645265}{25861781} a^{2} + \frac{4911188}{25861781} a + \frac{8249167}{25861781}$

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:  $15$
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:  \( 7380532.4291 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 4.4.32625.1, 4.4.725.1, 8.8.47897578125.1 x2, 8.8.1064390625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
5Data not computed
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$