↑

Properties

Label 16.8.15712755557...5489.7
Degree $16$
Signature $[8, 4]$
Discriminant $23^{10}\cdot 41^{14}$
Root discriminant $182.92$
Ramified primes $23, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T817

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![445302103, -1343332342, 1437441882, -603931162, 32646687, 58692800, -39826634, 16756686, -4628026, 1239752, -245620, 42356, -5964, 396, -10, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 10*x^14 + 396*x^13 - 5964*x^12 + 42356*x^11 - 245620*x^10 + 1239752*x^9 - 4628026*x^8 + 16756686*x^7 - 39826634*x^6 + 58692800*x^5 + 32646687*x^4 - 603931162*x^3 + 1437441882*x^2 - 1343332342*x + 445302103)
 
gp: K = bnfinit(x^16 - 6*x^15 - 10*x^14 + 396*x^13 - 5964*x^12 + 42356*x^11 - 245620*x^10 + 1239752*x^9 - 4628026*x^8 + 16756686*x^7 - 39826634*x^6 + 58692800*x^5 + 32646687*x^4 - 603931162*x^3 + 1437441882*x^2 - 1343332342*x + 445302103, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 10 x^{14} + 396 x^{13} - 5964 x^{12} + 42356 x^{11} - 245620 x^{10} + 1239752 x^{9} - 4628026 x^{8} + 16756686 x^{7} - 39826634 x^{6} + 58692800 x^{5} + 32646687 x^{4} - 603931162 x^{3} + 1437441882 x^{2} - 1343332342 x + 445302103 \)

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:  \(1571275555715210001755383712793895489=23^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.92$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{32} a^{12} + \frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{3}{32} a^{9} + \frac{1}{32} a^{8} + \frac{1}{8} a^{7} - \frac{5}{32} a^{6} - \frac{1}{8} a^{5} + \frac{1}{32} a^{4} - \frac{1}{32} a^{3} - \frac{9}{32} a^{2} - \frac{5}{32} a - \frac{1}{32}$, $\frac{1}{32} a^{13} + \frac{1}{16} a^{11} - \frac{1}{32} a^{10} + \frac{3}{32} a^{9} + \frac{1}{16} a^{8} + \frac{3}{32} a^{7} + \frac{3}{16} a^{6} + \frac{1}{32} a^{5} + \frac{5}{32} a^{4} + \frac{9}{32} a^{3} + \frac{5}{32} a^{2} + \frac{1}{32} a + \frac{1}{16}$, $\frac{1}{32} a^{14} + \frac{3}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{8} a^{9} + \frac{1}{32} a^{8} - \frac{1}{16} a^{7} - \frac{5}{32} a^{6} + \frac{5}{32} a^{5} - \frac{1}{32} a^{4} - \frac{1}{32} a^{3} + \frac{11}{32} a^{2} + \frac{3}{8} a + \frac{1}{16}$, $\frac{1}{1363278043364142470979704704578960349705600050616602976} a^{15} + \frac{614183395924200287533217384075502175621305865723389}{340819510841035617744926176144740087426400012654150744} a^{14} + \frac{810213194538570330814602151537508347724124186450075}{1363278043364142470979704704578960349705600050616602976} a^{13} - \frac{249116056167437001352286873285111860273312011856417}{42602438855129452218115772018092510928300001581768843} a^{12} + \frac{164203223665124291547636705790104107592585808075847955}{1363278043364142470979704704578960349705600050616602976} a^{11} + \frac{56331780691461152668527229036177865389980451235213947}{1363278043364142470979704704578960349705600050616602976} a^{10} + \frac{87940576071572447053675449227286972625011157851715617}{1363278043364142470979704704578960349705600050616602976} a^{9} - \frac{113554949765009377741144439739643453129055116075556099}{1363278043364142470979704704578960349705600050616602976} a^{8} - \frac{35022733570931088069119060815270045376522062043327875}{170409755420517808872463088072370043713200006327075372} a^{7} + \frac{21043441643714369707707053334571577608376752190308977}{681639021682071235489852352289480174852800025308301488} a^{6} + \frac{35148075094101501447932268983395993860922986091526857}{681639021682071235489852352289480174852800025308301488} a^{5} + \frac{79400853320806420966269411856958928395098085324595127}{1363278043364142470979704704578960349705600050616602976} a^{4} + \frac{412442594958089955246420596856549546144640991505463341}{1363278043364142470979704704578960349705600050616602976} a^{3} - \frac{84838130615401851026567875686750189146665939359980275}{681639021682071235489852352289480174852800025308301488} a^{2} - \frac{81476053039559211465665673335375458457566796914507703}{340819510841035617744926176144740087426400012654150744} a - \frac{5590240606286710276160832176080110619658300311059}{1220481686091443572945125071243473903048880976380128}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 657786528118 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$