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Properties

Label 16.0.21521298978...9664.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 17^{6}$
Root discriminant $18.66$
Ramified primes $2, 3, 17$
Class number $1$
Class group Trivial
Galois group $C_8:C_2^2$ (as 16T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 7, -30, -10, 72, 7, -156, 217, -168, 89, -48, 38, -30, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 17*x^14 - 30*x^13 + 38*x^12 - 48*x^11 + 89*x^10 - 168*x^9 + 217*x^8 - 156*x^7 + 7*x^6 + 72*x^5 - 10*x^4 - 30*x^3 + 7*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 17*x^14 - 30*x^13 + 38*x^12 - 48*x^11 + 89*x^10 - 168*x^9 + 217*x^8 - 156*x^7 + 7*x^6 + 72*x^5 - 10*x^4 - 30*x^3 + 7*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 17 x^{14} - 30 x^{13} + 38 x^{12} - 48 x^{11} + 89 x^{10} - 168 x^{9} + 217 x^{8} - 156 x^{7} + 7 x^{6} + 72 x^{5} - 10 x^{4} - 30 x^{3} + 7 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(215212989780710129664=2^{24}\cdot 3^{12}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.66$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} + \frac{6}{19} a^{13} + \frac{3}{19} a^{12} + \frac{3}{19} a^{11} + \frac{6}{19} a^{10} - \frac{6}{19} a^{9} - \frac{5}{19} a^{8} + \frac{3}{19} a^{7} - \frac{1}{19} a^{6} - \frac{8}{19} a^{5} - \frac{3}{19} a^{4} + \frac{2}{19} a^{3} + \frac{6}{19} a^{2} + \frac{3}{19} a + \frac{2}{19}$, $\frac{1}{301670467} a^{15} - \frac{6904906}{301670467} a^{14} + \frac{65340651}{301670467} a^{13} + \frac{84084023}{301670467} a^{12} - \frac{139754385}{301670467} a^{11} - \frac{105358208}{301670467} a^{10} + \frac{143954740}{301670467} a^{9} + \frac{6241125}{301670467} a^{8} + \frac{139577889}{301670467} a^{7} - \frac{1031992}{15877393} a^{6} + \frac{31866424}{301670467} a^{5} + \frac{29958808}{301670467} a^{4} + \frac{137708684}{301670467} a^{3} + \frac{95878401}{301670467} a^{2} + \frac{88780385}{301670467} a + \frac{17121948}{301670467}$

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:  \( -\frac{4550566}{15877393} a^{15} + \frac{28503726}{15877393} a^{14} - \frac{85258672}{15877393} a^{13} + \frac{159814846}{15877393} a^{12} - \frac{215546668}{15877393} a^{11} + \frac{274171929}{15877393} a^{10} - \frac{476146504}{15877393} a^{9} + \frac{892881222}{15877393} a^{8} - \frac{1230524264}{15877393} a^{7} + \frac{1033812086}{15877393} a^{6} - \frac{288813754}{15877393} a^{5} - \frac{266887560}{15877393} a^{4} + \frac{111654460}{15877393} a^{3} + \frac{146214876}{15877393} a^{2} - \frac{69712404}{15877393} a - \frac{11217209}{15877393} \) (order $6$)
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:  \( 13426.5980282 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T38):

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 $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), 4.4.9792.1, 4.0.1088.2, \(\Q(\sqrt{2}, \sqrt{-3})\), 8.4.14670139392.1, 8.4.14670139392.2, 8.0.95883264.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
3Data not computed
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$