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Properties

Label 16.0.14650433026...5376.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{46}\cdot 113^{6}$
Root discriminant $43.19$
Ramified primes $2, 113$
Class number $36$ (GRH)
Class group $[2, 18]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T330)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -112, 1508, -6032, 14797, -22620, 23058, -13324, 8433, -1800, 1668, -72, 57, -4, 18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 18*x^14 - 4*x^13 + 57*x^12 - 72*x^11 + 1668*x^10 - 1800*x^9 + 8433*x^8 - 13324*x^7 + 23058*x^6 - 22620*x^5 + 14797*x^4 - 6032*x^3 + 1508*x^2 - 112*x + 4)
 
gp: K = bnfinit(x^16 - 4*x^15 + 18*x^14 - 4*x^13 + 57*x^12 - 72*x^11 + 1668*x^10 - 1800*x^9 + 8433*x^8 - 13324*x^7 + 23058*x^6 - 22620*x^5 + 14797*x^4 - 6032*x^3 + 1508*x^2 - 112*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 18 x^{14} - 4 x^{13} + 57 x^{12} - 72 x^{11} + 1668 x^{10} - 1800 x^{9} + 8433 x^{8} - 13324 x^{7} + 23058 x^{6} - 22620 x^{5} + 14797 x^{4} - 6032 x^{3} + 1508 x^{2} - 112 x + 4 \)

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:  \(146504330269581929271525376=2^{46}\cdot 113^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.19$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{376} a^{14} + \frac{35}{376} a^{13} - \frac{31}{376} a^{12} + \frac{57}{376} a^{11} - \frac{5}{47} a^{10} + \frac{57}{188} a^{9} + \frac{41}{94} a^{8} - \frac{23}{47} a^{7} + \frac{37}{376} a^{6} + \frac{135}{376} a^{5} + \frac{69}{376} a^{4} + \frac{117}{376} a^{3} + \frac{12}{47} a^{2} - \frac{15}{188} a - \frac{35}{94}$, $\frac{1}{1378543065734755627576771208} a^{15} - \frac{62616307152232100208425}{1378543065734755627576771208} a^{14} + \frac{158524722037709907338502541}{1378543065734755627576771208} a^{13} - \frac{89067371083763130079485}{33623001603286722623823688} a^{12} + \frac{59850359449097111898455539}{344635766433688906894192802} a^{11} - \frac{83008365217082765743325375}{344635766433688906894192802} a^{10} + \frac{99695099324467635910285537}{344635766433688906894192802} a^{9} - \frac{132768147230076979776792581}{344635766433688906894192802} a^{8} - \frac{647449766732471547053355395}{1378543065734755627576771208} a^{7} + \frac{389221353058760254931678259}{1378543065734755627576771208} a^{6} + \frac{163053240741051583438194841}{1378543065734755627576771208} a^{5} + \frac{642067949966762525464949935}{1378543065734755627576771208} a^{4} - \frac{108466571910580636394192065}{344635766433688906894192802} a^{3} + \frac{68122579156211852710301089}{344635766433688906894192802} a^{2} - \frac{112208654312354729927869905}{344635766433688906894192802} a - \frac{44608409896626375404919101}{344635766433688906894192802}$

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

Class group and class number

$C_{2}\times C_{18}$, which has order $36$ (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:  \( 498795.173587 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T330):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7232.1, 8.0.6051948658688.13, 8.4.6694633472.2, 8.4.189123395584.6

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$113$113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$