↑

Properties

Label 16.0.81862996112...1809.5
Degree $16$
Signature $[0, 8]$
Discriminant $17^{10}\cdot 67^{8}$
Root discriminant $48.09$
Ramified primes $17, 67$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![69904, -96560, 37680, 6234, 2791, -6816, 2844, -2121, 140, 50, 195, 32, -32, 13, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 6*x^14 + 13*x^13 - 32*x^12 + 32*x^11 + 195*x^10 + 50*x^9 + 140*x^8 - 2121*x^7 + 2844*x^6 - 6816*x^5 + 2791*x^4 + 6234*x^3 + 37680*x^2 - 96560*x + 69904)
 
gp: K = bnfinit(x^16 - 2*x^15 - 6*x^14 + 13*x^13 - 32*x^12 + 32*x^11 + 195*x^10 + 50*x^9 + 140*x^8 - 2121*x^7 + 2844*x^6 - 6816*x^5 + 2791*x^4 + 6234*x^3 + 37680*x^2 - 96560*x + 69904, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 6 x^{14} + 13 x^{13} - 32 x^{12} + 32 x^{11} + 195 x^{10} + 50 x^{9} + 140 x^{8} - 2121 x^{7} + 2844 x^{6} - 6816 x^{5} + 2791 x^{4} + 6234 x^{3} + 37680 x^{2} - 96560 x + 69904 \)

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:  \(818629961123679547712831809=17^{10}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.09$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{268} a^{14} + \frac{3}{67} a^{13} + \frac{23}{134} a^{12} + \frac{69}{268} a^{11} - \frac{57}{134} a^{10} + \frac{20}{67} a^{9} + \frac{1}{4} a^{8} + \frac{4}{67} a^{7} + \frac{24}{67} a^{6} + \frac{47}{268} a^{5} - \frac{65}{134} a^{4} + \frac{29}{67} a^{3} - \frac{69}{268} a^{2} + \frac{30}{67} a - \frac{18}{67}$, $\frac{1}{19267179255166961644868609688536} a^{15} + \frac{1564381719950762471088913975}{4816794813791740411217152422134} a^{14} + \frac{2077113352241087828804378358911}{9633589627583480822434304844268} a^{13} - \frac{5920427971486248260305675527119}{19267179255166961644868609688536} a^{12} - \frac{1995550613371173662995781811181}{9633589627583480822434304844268} a^{11} - \frac{523749098782228396745172971400}{2408397406895870205608576211067} a^{10} - \frac{6712594547213764889040025091445}{19267179255166961644868609688536} a^{9} + \frac{2160112608675913334963118702895}{4816794813791740411217152422134} a^{8} - \frac{191499816595154821093338522349}{2408397406895870205608576211067} a^{7} - \frac{1767757949194779020518716691577}{19267179255166961644868609688536} a^{6} - \frac{3119472893775153974522468768753}{9633589627583480822434304844268} a^{5} + \frac{948420494508198409778750265648}{2408397406895870205608576211067} a^{4} + \frac{7226068352829366165474214719695}{19267179255166961644868609688536} a^{3} - \frac{2380968022807134125829504096583}{4816794813791740411217152422134} a^{2} - \frac{705326274046944026202360159685}{4816794813791740411217152422134} a + \frac{1067327035941942688422216495815}{2408397406895870205608576211067}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 2075822.13681 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-67}) \), 4.0.76313.1, 8.0.99002457473.2, 8.0.28611710209697.1, 8.0.1683041777041.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$