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Properties

Label 16.0.83220749810...0361.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{14}\cdot 23^{12}$
Root discriminant $85.61$
Ramified primes $11, 23$
Class number $3$ (GRH)
Class group $[3]$ (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![2808063, -9271368, 13401259, -11178929, 5788287, -1722505, 125405, 105894, -26783, -10055, 7483, -1719, 173, -25, 12, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 12*x^14 - 25*x^13 + 173*x^12 - 1719*x^11 + 7483*x^10 - 10055*x^9 - 26783*x^8 + 105894*x^7 + 125405*x^6 - 1722505*x^5 + 5788287*x^4 - 11178929*x^3 + 13401259*x^2 - 9271368*x + 2808063)
 
gp: K = bnfinit(x^16 - 2*x^15 + 12*x^14 - 25*x^13 + 173*x^12 - 1719*x^11 + 7483*x^10 - 10055*x^9 - 26783*x^8 + 105894*x^7 + 125405*x^6 - 1722505*x^5 + 5788287*x^4 - 11178929*x^3 + 13401259*x^2 - 9271368*x + 2808063, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 12 x^{14} - 25 x^{13} + 173 x^{12} - 1719 x^{11} + 7483 x^{10} - 10055 x^{9} - 26783 x^{8} + 105894 x^{7} + 125405 x^{6} - 1722505 x^{5} + 5788287 x^{4} - 11178929 x^{3} + 13401259 x^{2} - 9271368 x + 2808063 \)

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:  \(8322074981098944220712357040361=11^{14}\cdot 23^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.61$
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:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{51} a^{11} - \frac{5}{51} a^{10} - \frac{5}{51} a^{9} - \frac{2}{17} a^{8} + \frac{8}{17} a^{7} + \frac{4}{17} a^{6} + \frac{6}{17} a^{5} - \frac{7}{17} a^{4} + \frac{11}{51} a^{3} + \frac{2}{51} a^{2} + \frac{14}{51} a - \frac{7}{17}$, $\frac{1}{8211} a^{12} - \frac{2}{2737} a^{11} + \frac{6}{161} a^{10} + \frac{968}{8211} a^{9} - \frac{173}{1173} a^{8} + \frac{2861}{8211} a^{7} + \frac{2692}{8211} a^{6} - \frac{178}{1173} a^{5} + \frac{583}{2737} a^{4} - \frac{232}{1173} a^{3} - \frac{3422}{8211} a^{2} - \frac{1332}{2737} a - \frac{10}{2737}$, $\frac{1}{8211} a^{13} - \frac{52}{8211} a^{11} - \frac{1060}{8211} a^{10} + \frac{733}{8211} a^{9} + \frac{88}{2737} a^{8} + \frac{394}{2737} a^{7} - \frac{881}{2737} a^{6} + \frac{94}{357} a^{5} + \frac{649}{2737} a^{4} - \frac{3023}{8211} a^{3} - \frac{77}{1173} a^{2} - \frac{1144}{8211} a - \frac{543}{2737}$, $\frac{1}{2652153} a^{14} - \frac{41}{884051} a^{13} + \frac{29}{884051} a^{12} + \frac{6599}{884051} a^{11} + \frac{7111}{52003} a^{10} + \frac{258304}{2652153} a^{9} - \frac{22859}{378879} a^{8} - \frac{483544}{2652153} a^{7} + \frac{303204}{884051} a^{6} + \frac{412907}{2652153} a^{5} - \frac{139946}{2652153} a^{4} + \frac{601184}{2652153} a^{3} + \frac{266017}{2652153} a^{2} + \frac{575077}{2652153} a + \frac{210138}{884051}$, $\frac{1}{6050780849691169944328828089} a^{15} + \frac{152443167638485961687}{6050780849691169944328828089} a^{14} + \frac{236796933005966346694}{13846180434075903762766197} a^{13} + \frac{98541144188113995223057}{6050780849691169944328828089} a^{12} + \frac{4344573868688678944097812}{2016926949897056648109609363} a^{11} + \frac{29180060161978690147699624}{288132421413865235444229909} a^{10} - \frac{27000109077569575436230694}{864397264241595706332689727} a^{9} + \frac{97538346595541130747952145}{6050780849691169944328828089} a^{8} + \frac{476465541024094852321096507}{2016926949897056648109609363} a^{7} - \frac{8049391654250680356775177}{35384683331527309615958059} a^{6} + \frac{286035774586197838807567145}{6050780849691169944328828089} a^{5} - \frac{71236201492711161278645936}{318462149983745786543622531} a^{4} - \frac{471552398703948913169184710}{6050780849691169944328828089} a^{3} - \frac{107567951837987635833165442}{6050780849691169944328828089} a^{2} - \frac{461197834674914515797411070}{2016926949897056648109609363} a + \frac{97774942227191380507920950}{672308983299018882703203121}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -1 \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 297238964.584 \) (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

\(\Q(\sqrt{-23}) \), \(\Q(\sqrt{253}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{-23})\), 4.2.704099.1 x2, 4.0.30613.1 x2, 8.0.495755401801.1, 8.2.2884800683080019.1 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
11Data not computed
$23$23.8.6.1$x^{8} + 299 x^{4} + 25921$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
23.8.6.1$x^{8} + 299 x^{4} + 25921$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$