↑

Properties

Label 16.0.45892197269...7136.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 17^{6}\cdot 97^{6}$
Root discriminant $146.68$
Ramified primes $2, 3, 17, 97$
Class number $320$ (GRH)
Class group $[2, 2, 2, 40]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6079069441, 114227100, 1191614404, -129903372, 128950970, 11516316, 17814694, -27168, 238438, -112260, 32918, 2844, 1031, -216, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 216*x^13 + 1031*x^12 + 2844*x^11 + 32918*x^10 - 112260*x^9 + 238438*x^8 - 27168*x^7 + 17814694*x^6 + 11516316*x^5 + 128950970*x^4 - 129903372*x^3 + 1191614404*x^2 + 114227100*x + 6079069441)
 
gp: K = bnfinit(x^16 - 4*x^14 - 216*x^13 + 1031*x^12 + 2844*x^11 + 32918*x^10 - 112260*x^9 + 238438*x^8 - 27168*x^7 + 17814694*x^6 + 11516316*x^5 + 128950970*x^4 - 129903372*x^3 + 1191614404*x^2 + 114227100*x + 6079069441, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 216 x^{13} + 1031 x^{12} + 2844 x^{11} + 32918 x^{10} - 112260 x^{9} + 238438 x^{8} - 27168 x^{7} + 17814694 x^{6} + 11516316 x^{5} + 128950970 x^{4} - 129903372 x^{3} + 1191614404 x^{2} + 114227100 x + 6079069441 \)

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:  \(45892197269607041217434019101147136=2^{32}\cdot 3^{12}\cdot 17^{6}\cdot 97^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $146.68$
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:  $2, 3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} + \frac{6}{23} a^{13} - \frac{7}{23} a^{12} - \frac{9}{23} a^{11} + \frac{8}{23} a^{10} - \frac{8}{23} a^{8} + \frac{1}{23} a^{7} - \frac{7}{23} a^{6} + \frac{6}{23} a^{5} + \frac{8}{23} a^{4} + \frac{7}{23} a^{3} + \frac{4}{23} a^{2} + \frac{11}{23} a$, $\frac{1}{410243153871206182236327468025155328157540751393992835371179597151} a^{15} + \frac{7926795646035046520145457088461828519261154656630208873761469590}{410243153871206182236327468025155328157540751393992835371179597151} a^{14} + \frac{6044170682756764678505705170249543378283393727608315834999045944}{17836658863965486184188150783702405572066989191043166755268678137} a^{13} + \frac{102446708132168255423793301058274793163100601383047555829299885635}{410243153871206182236327468025155328157540751393992835371179597151} a^{12} + \frac{77908344912156705874232180641168004732729755932917104018885972383}{410243153871206182236327468025155328157540751393992835371179597151} a^{11} - \frac{169126135445033286093842519302505640517113171052414479682277700202}{410243153871206182236327468025155328157540751393992835371179597151} a^{10} - \frac{103444167620662905576377701833884443924934182993165017651237027087}{410243153871206182236327468025155328157540751393992835371179597151} a^{9} - \frac{118572632415597569388357291072630268910539751432028686439518212165}{410243153871206182236327468025155328157540751393992835371179597151} a^{8} + \frac{183761136335427141231435478486996301149294930758020961039965595640}{410243153871206182236327468025155328157540751393992835371179597151} a^{7} + \frac{100269963811796108222591093975652129373817403866008294450029432374}{410243153871206182236327468025155328157540751393992835371179597151} a^{6} - \frac{134778467718792267157493967364840856470349211580636005348980171035}{410243153871206182236327468025155328157540751393992835371179597151} a^{5} - \frac{114473108406220440659838570528770436684307771070155357035621205635}{410243153871206182236327468025155328157540751393992835371179597151} a^{4} + \frac{163518630519784749721739120222225301803988936871987012404274518739}{410243153871206182236327468025155328157540751393992835371179597151} a^{3} + \frac{79232406030779307602895916090572693948811357420588228645501922908}{410243153871206182236327468025155328157540751393992835371179597151} a^{2} - \frac{197190709551639876635127959010534822485695737801215817845472124774}{410243153871206182236327468025155328157540751393992835371179597151} a - \frac{173306752338681882212060451282754852050945747147611183080945216}{379503380084372046472088314546859693022701897681769505431248471}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{40}$, which has order $320$ (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:  \( 153483762.357 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.0.949824.1, 4.0.422144.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.14434650095616.3

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$