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Properties

Label 16.0.79659417600...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{4}\cdot 5^{12}\cdot 7^{4}$
Root discriminant $20.25$
Ramified primes $2, 3, 5, 7$
Class number $2$
Class group $[2]$
Galois group $C_2^3:(C_2\times C_4)$ (as 16T68)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -270, 657, -1332, 2197, -3028, 3389, -3166, 2540, -1734, 1047, -536, 237, -88, 27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 27*x^14 - 88*x^13 + 237*x^12 - 536*x^11 + 1047*x^10 - 1734*x^9 + 2540*x^8 - 3166*x^7 + 3389*x^6 - 3028*x^5 + 2197*x^4 - 1332*x^3 + 657*x^2 - 270*x + 81)
 
gp: K = bnfinit(x^16 - 6*x^15 + 27*x^14 - 88*x^13 + 237*x^12 - 536*x^11 + 1047*x^10 - 1734*x^9 + 2540*x^8 - 3166*x^7 + 3389*x^6 - 3028*x^5 + 2197*x^4 - 1332*x^3 + 657*x^2 - 270*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 27 x^{14} - 88 x^{13} + 237 x^{12} - 536 x^{11} + 1047 x^{10} - 1734 x^{9} + 2540 x^{8} - 3166 x^{7} + 3389 x^{6} - 3028 x^{5} + 2197 x^{4} - 1332 x^{3} + 657 x^{2} - 270 x + 81 \)

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:  \(796594176000000000000=2^{24}\cdot 3^{4}\cdot 5^{12}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.25$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{12} a^{13} - \frac{1}{4} a^{11} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{12} a^{3} - \frac{1}{3} a^{2} + \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{72} a^{14} - \frac{2}{9} a^{11} - \frac{1}{24} a^{10} + \frac{11}{36} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} - \frac{2}{9} a^{6} - \frac{11}{36} a^{5} + \frac{17}{72} a^{4} - \frac{7}{18} a^{3} - \frac{7}{36} a^{2} - \frac{1}{6} a + \frac{3}{8}$, $\frac{1}{13842974990664} a^{15} - \frac{6635717597}{2307162498444} a^{14} - \frac{10602328993}{384527083074} a^{13} - \frac{567101811941}{6921487495332} a^{12} + \frac{105343433089}{1538108332296} a^{11} - \frac{1255387156225}{6921487495332} a^{10} + \frac{266617308421}{576790624611} a^{9} - \frac{148796632331}{1153581249222} a^{8} + \frac{1621477869053}{3460743747666} a^{7} - \frac{444207741737}{6921487495332} a^{6} + \frac{2600474901689}{13842974990664} a^{5} + \frac{3069338969017}{6921487495332} a^{4} - \frac{1195683740137}{6921487495332} a^{3} - \frac{586496761645}{2307162498444} a^{2} - \frac{602822521609}{1538108332296} a - \frac{31553762529}{128175694358}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{32487897515}{1153581249222} a^{15} + \frac{75076635159}{512702777432} a^{14} - \frac{121880637320}{192263541537} a^{13} + \frac{1106972523877}{576790624611} a^{12} - \frac{1889343387739}{384527083074} a^{11} + \frac{48227899611055}{4614324996888} a^{10} - \frac{4940307063479}{256351388716} a^{9} + \frac{3762826995515}{128175694358} a^{8} - \frac{23319443195411}{576790624611} a^{7} + \frac{25870515748651}{576790624611} a^{6} - \frac{98925363016679}{2307162498444} a^{5} + \frac{146061590337815}{4614324996888} a^{4} - \frac{20789868366071}{1153581249222} a^{3} + \frac{2090026575819}{256351388716} a^{2} - \frac{673302244928}{192263541537} a + \frac{633577471313}{512702777432} \) (order $10$)
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:  \( 15702.1468537 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:(C_2\times C_4)$ (as 16T68):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 34 conjugacy class representatives for $C_2^3:(C_2\times C_4)$
Character table for $C_2^3:(C_2\times C_4)$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.28224000000.1, 8.0.64000000.2, 8.4.1128960000.1

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 R R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$