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Properties

Label 16.0.18682108719...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 61^{6}\cdot 97^{4}$
Root discriminant $138.66$
Ramified primes $2, 5, 61, 97$
Class number $1053824$ (GRH)
Class group $[2, 2, 4, 65864]$ (GRH)
Galois group 16T864

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18098212481, 11160522222, 5715260614, 4162184006, 1758527272, 413636482, 222224758, 19606386, 15396726, 369154, 637218, -4102, 15312, -226, 194, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 194*x^14 - 226*x^13 + 15312*x^12 - 4102*x^11 + 637218*x^10 + 369154*x^9 + 15396726*x^8 + 19606386*x^7 + 222224758*x^6 + 413636482*x^5 + 1758527272*x^4 + 4162184006*x^3 + 5715260614*x^2 + 11160522222*x + 18098212481)
 
gp: K = bnfinit(x^16 - 2*x^15 + 194*x^14 - 226*x^13 + 15312*x^12 - 4102*x^11 + 637218*x^10 + 369154*x^9 + 15396726*x^8 + 19606386*x^7 + 222224758*x^6 + 413636482*x^5 + 1758527272*x^4 + 4162184006*x^3 + 5715260614*x^2 + 11160522222*x + 18098212481, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 194 x^{14} - 226 x^{13} + 15312 x^{12} - 4102 x^{11} + 637218 x^{10} + 369154 x^{9} + 15396726 x^{8} + 19606386 x^{7} + 222224758 x^{6} + 413636482 x^{5} + 1758527272 x^{4} + 4162184006 x^{3} + 5715260614 x^{2} + 11160522222 x + 18098212481 \)

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:  \(18682108719227553550336000000000000=2^{24}\cdot 5^{12}\cdot 61^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $138.66$
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:  $2, 5, 61, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{5}{12} a^{3} - \frac{1}{2} a^{2} + \frac{1}{12} a - \frac{5}{12}$, $\frac{1}{120} a^{12} - \frac{1}{15} a^{9} - \frac{1}{24} a^{8} - \frac{1}{6} a^{7} - \frac{2}{15} a^{6} + \frac{1}{6} a^{5} - \frac{5}{24} a^{4} - \frac{13}{30} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{59}{120}$, $\frac{1}{120} a^{13} + \frac{1}{60} a^{10} - \frac{1}{8} a^{9} + \frac{1}{12} a^{8} + \frac{1}{30} a^{7} - \frac{1}{6} a^{6} - \frac{5}{24} a^{5} + \frac{1}{15} a^{4} - \frac{1}{2} a^{3} + \frac{1}{12} a^{2} + \frac{31}{120} a - \frac{5}{12}$, $\frac{1}{2280} a^{14} - \frac{1}{380} a^{13} + \frac{1}{1140} a^{12} - \frac{13}{380} a^{11} - \frac{9}{760} a^{10} - \frac{32}{285} a^{9} + \frac{41}{570} a^{8} + \frac{109}{570} a^{7} - \frac{517}{2280} a^{6} + \frac{23}{380} a^{5} + \frac{77}{380} a^{4} - \frac{427}{1140} a^{3} - \frac{3}{760} a^{2} - \frac{149}{570} a - \frac{1}{10}$, $\frac{1}{6357172291755003086916178961936831254219601498915813413863720} a^{15} + \frac{1183152144460147656346228550482778879332855101198639598909}{6357172291755003086916178961936831254219601498915813413863720} a^{14} - \frac{3362346437677681291042049565247827749404095798246175950479}{1271434458351000617383235792387366250843920299783162682772744} a^{13} + \frac{937701078326156556835092731887919909786567474082900495303}{2119057430585001028972059653978943751406533832971937804621240} a^{12} + \frac{75462689769738135283174115092770957786200853206399649869881}{2119057430585001028972059653978943751406533832971937804621240} a^{11} + \frac{51234644337694222582589816214927685209081160365341573248945}{1271434458351000617383235792387366250843920299783162682772744} a^{10} - \frac{463561593124915612809746249310701638743259003827345899374807}{6357172291755003086916178961936831254219601498915813413863720} a^{9} + \frac{510115545434711908821336792129341323672637460208665238004131}{6357172291755003086916178961936831254219601498915813413863720} a^{8} - \frac{106741231618060120668054542076372796345637600562570938306257}{1271434458351000617383235792387366250843920299783162682772744} a^{7} + \frac{1137083448015508750664801800671304595715229624963657851297451}{6357172291755003086916178961936831254219601498915813413863720} a^{6} - \frac{826271007094554768787017594811841315896667150847562860350893}{6357172291755003086916178961936831254219601498915813413863720} a^{5} - \frac{71285067774218550173403094828035274534985813452319350912463}{423811486117000205794411930795788750281306766594387560924248} a^{4} + \frac{837901145561690530207093541209992523793074248351459214671299}{2119057430585001028972059653978943751406533832971937804621240} a^{3} - \frac{1589309299367591826015705314885183098341627963256796908000021}{6357172291755003086916178961936831254219601498915813413863720} a^{2} - \frac{492377366589384575928407724753198804935655103452274066302077}{1271434458351000617383235792387366250843920299783162682772744} a - \frac{80029789866208303306761802830751143430010042862982586022437}{334588015355526478258746261154570066011557973627148074413880}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{65864}$, which has order $1053824$ (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:  \( 26657.4092537 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T864:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n864
Character table for t16n864 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.14884000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61Data not computed
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$