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Properties

Label 16.0.18910794979...2384.9
Degree $16$
Signature $[0, 8]$
Discriminant $2^{58}\cdot 3^{8}$
Root discriminant $21.37$
Ramified primes $2, 3$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![9604, -14112, 10368, -2016, 10192, -14400, 10368, -1152, -948, 720, 0, -144, 40, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^12 - 144*x^11 + 720*x^9 - 948*x^8 - 1152*x^7 + 10368*x^6 - 14400*x^5 + 10192*x^4 - 2016*x^3 + 10368*x^2 - 14112*x + 9604)
 
gp: K = bnfinit(x^16 + 40*x^12 - 144*x^11 + 720*x^9 - 948*x^8 - 1152*x^7 + 10368*x^6 - 14400*x^5 + 10192*x^4 - 2016*x^3 + 10368*x^2 - 14112*x + 9604, 1)
 

Normalized defining polynomial

\( x^{16} + 40 x^{12} - 144 x^{11} + 720 x^{9} - 948 x^{8} - 1152 x^{7} + 10368 x^{6} - 14400 x^{5} + 10192 x^{4} - 2016 x^{3} + 10368 x^{2} - 14112 x + 9604 \)

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:  \(1891079497931380752384=2^{58}\cdot 3^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.37$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{24} a^{8} - \frac{1}{2} a^{6} - \frac{1}{6} a^{4} - \frac{1}{12}$, $\frac{1}{24} a^{9} - \frac{1}{2} a^{7} - \frac{1}{6} a^{5} - \frac{1}{12} a$, $\frac{1}{24} a^{10} - \frac{1}{6} a^{6} - \frac{1}{12} a^{2}$, $\frac{1}{48} a^{11} + \frac{5}{12} a^{7} - \frac{1}{2} a^{5} - \frac{1}{24} a^{3} - \frac{1}{2} a$, $\frac{1}{48} a^{12} - \frac{1}{2} a^{6} - \frac{3}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{6}$, $\frac{1}{48} a^{13} - \frac{1}{2} a^{7} - \frac{3}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{6} a$, $\frac{1}{238479537072} a^{14} - \frac{303134215}{59619884268} a^{13} - \frac{1516947829}{238479537072} a^{12} + \frac{27271}{2655408} a^{11} - \frac{1032240653}{59619884268} a^{10} - \frac{399519809}{29809942134} a^{9} + \frac{175947367}{119239768536} a^{8} - \frac{1970453179}{8517126324} a^{7} + \frac{40025730607}{119239768536} a^{6} + \frac{7105544830}{14904971067} a^{5} + \frac{1970602427}{17034252648} a^{4} - \frac{45637696255}{119239768536} a^{3} - \frac{10659559271}{29809942134} a^{2} - \frac{6196007701}{29809942134} a + \frac{93813667}{1216732332}$, $\frac{1}{291660473839056} a^{15} - \frac{99}{48610078973176} a^{14} + \frac{241629791829}{97220157946352} a^{13} - \frac{67350747071}{18228779614941} a^{12} + \frac{895842757379}{145830236919528} a^{11} + \frac{500687748433}{36457559229882} a^{10} - \frac{1422704572855}{72915118459764} a^{9} + \frac{494169815261}{24305039486588} a^{8} - \frac{43804287364409}{145830236919528} a^{7} - \frac{112979375951}{1488063642036} a^{6} + \frac{46489110597653}{145830236919528} a^{5} + \frac{1134758369823}{6076259871647} a^{4} - \frac{20787901897303}{72915118459764} a^{3} + \frac{4102976717978}{18228779614941} a^{2} + \frac{1702379054099}{18228779614941} a - \frac{62125246129}{744031821018}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

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:  \( -\frac{275821129}{6076259871647} a^{15} - \frac{27873714825}{97220157946352} a^{14} - \frac{5372734779}{12152519743294} a^{13} - \frac{11138615613}{24305039486588} a^{12} - \frac{13143528999}{6944296996168} a^{11} - \frac{15008267457}{3472148498084} a^{10} + \frac{1197377203005}{48610078973176} a^{9} + \frac{303030039213}{24305039486588} a^{8} - \frac{625714790958}{6076259871647} a^{7} + \frac{2203758372321}{48610078973176} a^{6} - \frac{929974797783}{12152519743294} a^{5} - \frac{9516214311804}{6076259871647} a^{4} - \frac{887651282711}{24305039486588} a^{3} - \frac{4172806612944}{6076259871647} a^{2} + \frac{89805648591}{24305039486588} a - \frac{542744779329}{248010607006} \) (order $8$)
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 25045.3091327 \) (assuming GRH)
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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{8})\), 4.2.1024.1 x2, 4.0.512.1 x2, 8.0.4194304.1, 8.2.21743271936.2 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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$