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Properties

Label 16.0.23595621172...7056.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 11^{8}$
Root discriminant $16.25$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $D_4\times C_2$ (as 16T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -72, 288, -732, 1270, -1500, 1128, -364, -217, 308, -88, -100, 138, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 138*x^12 - 100*x^11 - 88*x^10 + 308*x^9 - 217*x^8 - 364*x^7 + 1128*x^6 - 1500*x^5 + 1270*x^4 - 732*x^3 + 288*x^2 - 72*x + 9)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 138*x^12 - 100*x^11 - 88*x^10 + 308*x^9 - 217*x^8 - 364*x^7 + 1128*x^6 - 1500*x^5 + 1270*x^4 - 732*x^3 + 288*x^2 - 72*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 138 x^{12} - 100 x^{11} - 88 x^{10} + 308 x^{9} - 217 x^{8} - 364 x^{7} + 1128 x^{6} - 1500 x^{5} + 1270 x^{4} - 732 x^{3} + 288 x^{2} - 72 x + 9 \)

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:  \(23595621172490797056=2^{24}\cdot 3^{8}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.25$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{12} a^{12} + \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} + \frac{1}{6} a^{3} + \frac{1}{12} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{7} - \frac{1}{4} a^{6} + \frac{1}{6} a^{4} + \frac{1}{12} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{3000} a^{14} - \frac{7}{3000} a^{13} - \frac{3}{125} a^{12} - \frac{227}{3000} a^{11} - \frac{7}{200} a^{10} + \frac{8}{375} a^{9} + \frac{161}{3000} a^{8} - \frac{239}{3000} a^{7} - \frac{323}{3000} a^{6} + \frac{373}{1500} a^{5} + \frac{241}{600} a^{4} + \frac{281}{1000} a^{3} + \frac{739}{1500} a^{2} + \frac{3}{40} a - \frac{451}{1000}$, $\frac{1}{501000} a^{15} + \frac{19}{125250} a^{14} + \frac{13597}{501000} a^{13} + \frac{2099}{167000} a^{12} + \frac{13451}{125250} a^{11} + \frac{16783}{167000} a^{10} - \frac{31777}{501000} a^{9} + \frac{1687}{250500} a^{8} + \frac{2053}{16700} a^{7} + \frac{33437}{501000} a^{6} - \frac{59209}{167000} a^{5} + \frac{19777}{125250} a^{4} + \frac{75899}{167000} a^{3} - \frac{40367}{167000} a^{2} - \frac{23863}{83500} a - \frac{5183}{167000}$

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:  \( -\frac{40363}{250500} a^{15} + \frac{683267}{501000} a^{14} - \frac{943341}{167000} a^{13} + \frac{317718}{20875} a^{12} - \frac{870701}{33400} a^{11} + \frac{3467537}{167000} a^{10} + \frac{584017}{41750} a^{9} - \frac{28930451}{501000} a^{8} + \frac{24077533}{501000} a^{7} + \frac{10083883}{167000} a^{6} - \frac{10493257}{50100} a^{5} + \frac{48554719}{167000} a^{4} - \frac{41509441}{167000} a^{3} + \frac{1754252}{12525} a^{2} - \frac{8323449}{167000} a + \frac{318213}{33400} \) (order $12$)
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:  \( 11716.986321 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(i, \sqrt{33})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{11})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{11})\), 4.2.6336.2 x2, 4.2.6336.1 x2, 4.0.5808.1 x2, 4.0.5808.2 x2, 8.0.303595776.1, 8.0.40144896.2 x2, 8.0.539725824.2 x2, 8.4.4857532416.1 x2, 8.0.303595776.3 x2, 8.0.4857532416.5, 8.0.4857532416.7

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$