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Properties

Label 16.8.64557605582...0625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 13^{2}\cdot 97^{4}\cdot 101^{10}$
Root discriminant $173.03$
Ramified primes $5, 13, 97, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T860

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18802865920, 27785101440, -2403534464, -1784768672, -1273883888, 46417704, -27498496, 20224282, 8042475, -1573763, 469183, -64746, -3859, 212, -186, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 186*x^14 + 212*x^13 - 3859*x^12 - 64746*x^11 + 469183*x^10 - 1573763*x^9 + 8042475*x^8 + 20224282*x^7 - 27498496*x^6 + 46417704*x^5 - 1273883888*x^4 - 1784768672*x^3 - 2403534464*x^2 + 27785101440*x - 18802865920)
 
gp: K = bnfinit(x^16 - 2*x^15 - 186*x^14 + 212*x^13 - 3859*x^12 - 64746*x^11 + 469183*x^10 - 1573763*x^9 + 8042475*x^8 + 20224282*x^7 - 27498496*x^6 + 46417704*x^5 - 1273883888*x^4 - 1784768672*x^3 - 2403534464*x^2 + 27785101440*x - 18802865920, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 186 x^{14} + 212 x^{13} - 3859 x^{12} - 64746 x^{11} + 469183 x^{10} - 1573763 x^{9} + 8042475 x^{8} + 20224282 x^{7} - 27498496 x^{6} + 46417704 x^{5} - 1273883888 x^{4} - 1784768672 x^{3} - 2403534464 x^{2} + 27785101440 x - 18802865920 \)

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:  \(645576055826149774217152338862890625=5^{8}\cdot 13^{2}\cdot 97^{4}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $173.03$
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:  $5, 13, 97, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{3}{8} a^{7} - \frac{1}{8} a^{5} + \frac{3}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{3}{16} a^{8} + \frac{7}{16} a^{6} - \frac{5}{16} a^{5} - \frac{7}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{13} - \frac{1}{16} a^{11} - \frac{3}{32} a^{9} - \frac{1}{2} a^{8} - \frac{9}{32} a^{7} - \frac{5}{32} a^{6} + \frac{9}{32} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{14} - \frac{1}{32} a^{12} - \frac{3}{64} a^{10} - \frac{1}{4} a^{9} - \frac{9}{64} a^{8} + \frac{27}{64} a^{7} - \frac{23}{64} a^{6} - \frac{7}{16} a^{5} + \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{3738932770054065389877241011370614296789834265552698811342520751394106306688} a^{15} - \frac{4347033672318581864147773067308020634038697603668105456651629678691272507}{934733192513516347469310252842653574197458566388174702835630187848526576672} a^{14} + \frac{39141145652949813598219167099545924581194715112111603848563249885682439}{5860396191307312523318559578950806107821056842559089045991411836040918976} a^{13} - \frac{1569843303307283037713075482939643421850712974035629661551285063668556539}{58420824532094771716831890802665848387341160399260918927226886740532911042} a^{12} - \frac{230547032641834112192102639714921092029815683946432677439543475638392033579}{3738932770054065389877241011370614296789834265552698811342520751394106306688} a^{11} - \frac{87031627669249644850989406027753202715747760136715660700114263103882349047}{934733192513516347469310252842653574197458566388174702835630187848526576672} a^{10} - \frac{283806473121951039227179389720170874396545604375186203215278960732440734373}{3738932770054065389877241011370614296789834265552698811342520751394106306688} a^{9} - \frac{1776974666440136399453763553667379046433467055867956888295279216073557071569}{3738932770054065389877241011370614296789834265552698811342520751394106306688} a^{8} - \frac{1124688231003378975402879305911956095675361405853087207015243641224362822799}{3738932770054065389877241011370614296789834265552698811342520751394106306688} a^{7} - \frac{456035226579881965946070706255037722220626133451005620030177885131593504261}{934733192513516347469310252842653574197458566388174702835630187848526576672} a^{6} + \frac{31260688236311980296275761246990643246807104828172592881239840964349408529}{84975744773956031588119113894786688563405324217106791166875471622593325152} a^{5} - \frac{18012855181066751004252624602064508539432646149102052887323748862303000979}{233683298128379086867327563210663393549364641597043675708907546962131644168} a^{4} - \frac{9538283461625399565512592764802510631615322467419596418629774523728283374}{29210412266047385858415945401332924193670580199630459463613443370266455521} a^{3} - \frac{7855474677940576471156232035090457022024017295654316427591584191913826288}{29210412266047385858415945401332924193670580199630459463613443370266455521} a^{2} + \frac{10864369479957729045341935783402575039018445434483306120978625456393243419}{29210412266047385858415945401332924193670580199630459463613443370266455521} a + \frac{6942764018123514532040647803407122009298972475121377569825168495496264317}{29210412266047385858415945401332924193670580199630459463613443370266455521}$

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:  $11$
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:  \( 338676992490 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T860:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n860 are not computed
Character table for t16n860 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{505}) \), 4.4.51005.1 x2, 4.4.2525.1 x2, \(\Q(\sqrt{5}, \sqrt{101})\), 8.8.65037750625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{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$$[\ ]_{2}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
101Data not computed