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Properties

Label 16.0.10770650185...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{14}\cdot 5^{8}\cdot 7^{8}$
Root discriminant $15.47$
Ramified primes $3, 5, 7$
Class number $2$
Class group $[2]$
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![9, -27, 54, -72, 120, -144, 117, -84, 79, -59, 22, -2, 10, -17, 13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 13*x^14 - 17*x^13 + 10*x^12 - 2*x^11 + 22*x^10 - 59*x^9 + 79*x^8 - 84*x^7 + 117*x^6 - 144*x^5 + 120*x^4 - 72*x^3 + 54*x^2 - 27*x + 9)
 
gp: K = bnfinit(x^16 - 5*x^15 + 13*x^14 - 17*x^13 + 10*x^12 - 2*x^11 + 22*x^10 - 59*x^9 + 79*x^8 - 84*x^7 + 117*x^6 - 144*x^5 + 120*x^4 - 72*x^3 + 54*x^2 - 27*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 13 x^{14} - 17 x^{13} + 10 x^{12} - 2 x^{11} + 22 x^{10} - 59 x^{9} + 79 x^{8} - 84 x^{7} + 117 x^{6} - 144 x^{5} + 120 x^{4} - 72 x^{3} + 54 x^{2} - 27 x + 9 \)

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:  \(10770650185222265625=3^{14}\cdot 5^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.47$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{4}$, $\frac{1}{15} a^{14} - \frac{1}{15} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} - \frac{1}{15} a^{5} - \frac{1}{3} a^{4} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{901399245} a^{15} + \frac{400679}{901399245} a^{14} - \frac{5578734}{60093283} a^{13} - \frac{113814743}{901399245} a^{12} + \frac{144585253}{901399245} a^{11} + \frac{7943617}{901399245} a^{10} - \frac{200873177}{901399245} a^{9} + \frac{22367777}{180279849} a^{8} - \frac{20625979}{53023485} a^{7} + \frac{132568148}{300466415} a^{6} - \frac{23636159}{180279849} a^{5} - \frac{73555864}{180279849} a^{4} - \frac{29147817}{60093283} a^{3} + \frac{27754671}{300466415} a^{2} - \frac{10356998}{300466415} a - \frac{4308502}{60093283}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 881.706230923 \)
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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-35}) \), 4.2.33075.1, 4.2.1323.1, \(\Q(\sqrt{-15}, \sqrt{21})\), 8.2.131274675.2, 8.2.131274675.1, 8.0.1093955625.2

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 R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
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}$