↑

Properties

Label 16.8.12161637217...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{14}\cdot 109^{8}$
Root discriminant $42.69$
Ramified primes $5, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -704, 1168, 3272, -16064, 7636, 29095, -31124, -3421, 13297, -2240, -1762, 446, 99, -33, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 33*x^14 + 99*x^13 + 446*x^12 - 1762*x^11 - 2240*x^10 + 13297*x^9 - 3421*x^8 - 31124*x^7 + 29095*x^6 + 7636*x^5 - 16064*x^4 + 3272*x^3 + 1168*x^2 - 704*x + 256)
 
gp: K = bnfinit(x^16 - 2*x^15 - 33*x^14 + 99*x^13 + 446*x^12 - 1762*x^11 - 2240*x^10 + 13297*x^9 - 3421*x^8 - 31124*x^7 + 29095*x^6 + 7636*x^5 - 16064*x^4 + 3272*x^3 + 1168*x^2 - 704*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 33 x^{14} + 99 x^{13} + 446 x^{12} - 1762 x^{11} - 2240 x^{10} + 13297 x^{9} - 3421 x^{8} - 31124 x^{7} + 29095 x^{6} + 7636 x^{5} - 16064 x^{4} + 3272 x^{3} + 1168 x^{2} - 704 x + 256 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(121616372173473638916015625=5^{14}\cdot 109^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.69$
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:  $5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{3}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{3}{16} a^{7} - \frac{1}{4} a^{6} - \frac{7}{16} a^{5} - \frac{3}{16} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{272} a^{14} - \frac{1}{34} a^{13} - \frac{7}{272} a^{12} + \frac{11}{272} a^{11} + \frac{3}{34} a^{10} - \frac{5}{34} a^{9} + \frac{11}{136} a^{8} + \frac{71}{272} a^{7} + \frac{123}{272} a^{6} + \frac{67}{136} a^{5} - \frac{3}{16} a^{4} - \frac{9}{34} a^{3} - \frac{8}{17} a^{2} + \frac{3}{17} a - \frac{7}{17}$, $\frac{1}{5735243172089373916764736} a^{15} - \frac{4501627733176292533873}{2867621586044686958382368} a^{14} - \frac{106302197634368473053149}{5735243172089373916764736} a^{13} - \frac{271237654520336375337969}{5735243172089373916764736} a^{12} - \frac{4066360246227506974085}{77503286109315863740064} a^{11} + \frac{29802009215167237811435}{260692871458607905307488} a^{10} - \frac{280575584100853377706237}{1433810793022343479191184} a^{9} + \frac{17574831676928331935157}{139883979807057900408896} a^{8} + \frac{129915839444867522683495}{337367245417021995103808} a^{7} + \frac{595680045793294682188475}{1433810793022343479191184} a^{6} + \frac{1755429841321007235954235}{5735243172089373916764736} a^{5} - \frac{11001953614225320234147}{65173217864651976326872} a^{4} + \frac{212927745669695871092569}{716905396511171739595592} a^{3} - \frac{117538408675624833948531}{716905396511171739595592} a^{2} + \frac{24322076087745899678621}{358452698255585869797796} a + \frac{29690998623265994931467}{89613174563896467449449}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 39488908.4077 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{545}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{109}) \), 4.4.1485125.1 x2, 4.4.13625.1 x2, \(\Q(\sqrt{5}, \sqrt{109})\), 8.4.11027981328125.2, 8.4.11027981328125.1, 8.8.2205596265625.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$109$109.8.4.1$x^{8} + 712860 x^{4} - 1295029 x^{2} + 127042344900$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
109.8.4.1$x^{8} + 712860 x^{4} - 1295029 x^{2} + 127042344900$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$