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Properties

Label 16.0.20181463130...0281.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 11^{14}$
Root discriminant $18.58$
Ramified primes $3, 11$
Class number $1$
Class group Trivial
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![144, -576, 1044, -1332, 1683, -2259, 2784, -2901, 2554, -1895, 1178, -623, 286, -109, 32, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 32*x^14 - 109*x^13 + 286*x^12 - 623*x^11 + 1178*x^10 - 1895*x^9 + 2554*x^8 - 2901*x^7 + 2784*x^6 - 2259*x^5 + 1683*x^4 - 1332*x^3 + 1044*x^2 - 576*x + 144)
 
gp: K = bnfinit(x^16 - 7*x^15 + 32*x^14 - 109*x^13 + 286*x^12 - 623*x^11 + 1178*x^10 - 1895*x^9 + 2554*x^8 - 2901*x^7 + 2784*x^6 - 2259*x^5 + 1683*x^4 - 1332*x^3 + 1044*x^2 - 576*x + 144, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 32 x^{14} - 109 x^{13} + 286 x^{12} - 623 x^{11} + 1178 x^{10} - 1895 x^{9} + 2554 x^{8} - 2901 x^{7} + 2784 x^{6} - 2259 x^{5} + 1683 x^{4} - 1332 x^{3} + 1044 x^{2} - 576 x + 144 \)

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:  \(201814631309311180281=3^{12}\cdot 11^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.58$
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:  $3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} + \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{5}{24} a^{6} - \frac{5}{24} a^{5} - \frac{5}{24} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{48} a^{13} + \frac{1}{48} a^{11} - \frac{1}{24} a^{10} - \frac{1}{16} a^{9} + \frac{1}{24} a^{8} - \frac{1}{16} a^{7} + \frac{1}{8} a^{6} + \frac{5}{48} a^{5} - \frac{1}{6} a^{4} - \frac{5}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{96} a^{14} - \frac{1}{96} a^{12} - \frac{1}{16} a^{11} + \frac{5}{96} a^{10} + \frac{1}{24} a^{9} - \frac{11}{96} a^{8} - \frac{1}{48} a^{7} + \frac{7}{96} a^{6} - \frac{1}{24} a^{5} + \frac{1}{96} a^{4} - \frac{5}{16} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{172133088} a^{15} + \frac{125133}{57377696} a^{14} - \frac{1549615}{172133088} a^{13} - \frac{239585}{172133088} a^{12} + \frac{6857521}{172133088} a^{11} + \frac{1261965}{57377696} a^{10} - \frac{7046687}{57377696} a^{9} - \frac{20888707}{172133088} a^{8} - \frac{9740341}{172133088} a^{7} + \frac{12043189}{172133088} a^{6} + \frac{11649251}{172133088} a^{5} + \frac{10587661}{172133088} a^{4} - \frac{1603701}{3586106} a^{3} - \frac{364511}{3586106} a^{2} - \frac{2863919}{7172212} a + \frac{33298}{1793053}$

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:  \( \frac{18865}{198768} a^{15} - \frac{233629}{397536} a^{14} + \frac{169037}{66256} a^{13} - \frac{1090441}{132512} a^{12} + \frac{503593}{24846} a^{11} - \frac{5577011}{132512} a^{10} + \frac{15167449}{198768} a^{9} - \frac{45821501}{397536} a^{8} + \frac{597256}{4141} a^{7} - \frac{60370607}{397536} a^{6} + \frac{8851249}{66256} a^{5} - \frac{13066955}{132512} a^{4} + \frac{4857611}{66256} a^{3} - \frac{257708}{4141} a^{2} + \frac{737187}{16564} a - \frac{120707}{8282} \) (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:  \( 53834.7389087 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-3}, \sqrt{-11})\), 4.2.11979.1 x2, 4.0.3993.1 x2, 8.0.143496441.1, 8.2.14206147659.1 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
11Data not computed