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Properties

Label 16.16.1022292086...5625.2
Degree $16$
Signature $[16, 0]$
Discriminant $5^{8}\cdot 29^{6}\cdot 89^{6}\cdot 97^{4}$
Root discriminant $133.54$
Ramified primes $5, 29, 89, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-413586661, 3700629245, -4125832615, -2964390016, 2583616864, 667980468, -466102134, -73288835, 35522224, 3954134, -1329612, -104119, 26010, 1289, -256, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 256*x^14 + 1289*x^13 + 26010*x^12 - 104119*x^11 - 1329612*x^10 + 3954134*x^9 + 35522224*x^8 - 73288835*x^7 - 466102134*x^6 + 667980468*x^5 + 2583616864*x^4 - 2964390016*x^3 - 4125832615*x^2 + 3700629245*x - 413586661)
 
gp: K = bnfinit(x^16 - 6*x^15 - 256*x^14 + 1289*x^13 + 26010*x^12 - 104119*x^11 - 1329612*x^10 + 3954134*x^9 + 35522224*x^8 - 73288835*x^7 - 466102134*x^6 + 667980468*x^5 + 2583616864*x^4 - 2964390016*x^3 - 4125832615*x^2 + 3700629245*x - 413586661, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 256 x^{14} + 1289 x^{13} + 26010 x^{12} - 104119 x^{11} - 1329612 x^{10} + 3954134 x^{9} + 35522224 x^{8} - 73288835 x^{7} - 466102134 x^{6} + 667980468 x^{5} + 2583616864 x^{4} - 2964390016 x^{3} - 4125832615 x^{2} + 3700629245 x - 413586661 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10222920866308585756291080141015625=5^{8}\cdot 29^{6}\cdot 89^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.54$
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:  $5, 29, 89, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{25} a^{12} + \frac{2}{25} a^{11} + \frac{2}{25} a^{10} + \frac{2}{25} a^{9} - \frac{6}{25} a^{7} - \frac{6}{25} a^{6} + \frac{12}{25} a^{5} - \frac{2}{5} a^{4} + \frac{2}{25} a^{3} + \frac{9}{25} a^{2} + \frac{12}{25} a - \frac{1}{25}$, $\frac{1}{25} a^{13} - \frac{2}{25} a^{11} - \frac{2}{25} a^{10} + \frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{4}{25} a^{6} - \frac{9}{25} a^{5} + \frac{7}{25} a^{4} - \frac{1}{5} a^{3} + \frac{9}{25} a^{2} - \frac{2}{5} a - \frac{3}{25}$, $\frac{1}{25} a^{14} + \frac{2}{25} a^{11} - \frac{2}{25} a^{9} + \frac{1}{25} a^{8} + \frac{12}{25} a^{7} + \frac{9}{25} a^{6} + \frac{11}{25} a^{5} - \frac{1}{5} a^{4} + \frac{3}{25} a^{3} - \frac{2}{25} a^{2} + \frac{6}{25} a + \frac{3}{25}$, $\frac{1}{106446825231196538131979769367988429328678569713184302298142325} a^{15} + \frac{150809058490033224984019784262937229011119068566374003103683}{106446825231196538131979769367988429328678569713184302298142325} a^{14} - \frac{1668469071556904937863472459747255067598693248983886894044844}{106446825231196538131979769367988429328678569713184302298142325} a^{13} - \frac{354122259019243517837968487204577693836035182504298272521257}{21289365046239307626395953873597685865735713942636860459628465} a^{12} + \frac{1589401829760877278751383819462859156846902006025248689941721}{21289365046239307626395953873597685865735713942636860459628465} a^{11} - \frac{6725718044063751184888083674227267869013625917323660489099573}{106446825231196538131979769367988429328678569713184302298142325} a^{10} + \frac{4769663218041805096696272818386983277734273694592305750106742}{106446825231196538131979769367988429328678569713184302298142325} a^{9} + \frac{78231052035104285692584449845878224008471975888455490924549}{106446825231196538131979769367988429328678569713184302298142325} a^{8} - \frac{18930049703490531218732821944665779193402581306276335094864212}{106446825231196538131979769367988429328678569713184302298142325} a^{7} + \frac{34870390088979220679506636904111985863539871074421331326349449}{106446825231196538131979769367988429328678569713184302298142325} a^{6} + \frac{8703498406271680707385668165466162171108292595364267904407674}{21289365046239307626395953873597685865735713942636860459628465} a^{5} + \frac{9337174083933191083834168738139789490153631531782976499540372}{21289365046239307626395953873597685865735713942636860459628465} a^{4} - \frac{46541946700182793551957431760442858731338916964939066761586922}{106446825231196538131979769367988429328678569713184302298142325} a^{3} + \frac{19126739683559402876025626882666665333565446769179332676729221}{106446825231196538131979769367988429328678569713184302298142325} a^{2} + \frac{51621887544024579612683464343163546917584209950546237882983547}{106446825231196538131979769367988429328678569713184302298142325} a + \frac{28887710490700741173200178280385559678173359345526914183389893}{106446825231196538131979769367988429328678569713184302298142325}$

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:  $15$
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:  \( 326411332294 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

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 $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.4.64525.1, 4.4.2225.1, 8.8.4163475625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.8.4.2$x^{8} - 912673 x^{2} + 2036173463$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$