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Properties

Label 16.8.10221605059...4917.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{9}\cdot 7^{14}\cdot 43^{8}\cdot 6551$
Root discriminant $115.64$
Ramified primes $3, 7, 43, 6551$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1638

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![87777, -820314, -4217589, 188181, -2441943, -200613, -56966, -102555, 30122, -15091, 5089, 1855, -602, 140, -19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 19*x^14 + 140*x^13 - 602*x^12 + 1855*x^11 + 5089*x^10 - 15091*x^9 + 30122*x^8 - 102555*x^7 - 56966*x^6 - 200613*x^5 - 2441943*x^4 + 188181*x^3 - 4217589*x^2 - 820314*x + 87777)
 
gp: K = bnfinit(x^16 - 6*x^15 - 19*x^14 + 140*x^13 - 602*x^12 + 1855*x^11 + 5089*x^10 - 15091*x^9 + 30122*x^8 - 102555*x^7 - 56966*x^6 - 200613*x^5 - 2441943*x^4 + 188181*x^3 - 4217589*x^2 - 820314*x + 87777, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 19 x^{14} + 140 x^{13} - 602 x^{12} + 1855 x^{11} + 5089 x^{10} - 15091 x^{9} + 30122 x^{8} - 102555 x^{7} - 56966 x^{6} - 200613 x^{5} - 2441943 x^{4} + 188181 x^{3} - 4217589 x^{2} - 820314 x + 87777 \)

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:  \(1022160505931823153325927974904917=3^{9}\cdot 7^{14}\cdot 43^{8}\cdot 6551\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.64$
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:  $3, 7, 43, 6551$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{21} a^{8} - \frac{10}{21} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{21} a^{9} + \frac{5}{21} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{4}{21} a^{2} + \frac{2}{7}$, $\frac{1}{21} a^{10} - \frac{2}{7} a^{7} + \frac{1}{3} a^{6} - \frac{1}{7} a^{3} + \frac{1}{3} a^{2} - \frac{1}{7}$, $\frac{1}{63} a^{11} + \frac{1}{63} a^{10} - \frac{1}{63} a^{9} - \frac{22}{63} a^{7} - \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{4}{63} a^{4} + \frac{4}{63} a^{3} - \frac{2}{7} a^{2} - \frac{1}{3} a - \frac{2}{7}$, $\frac{1}{441} a^{12} - \frac{1}{441} a^{11} + \frac{2}{147} a^{10} - \frac{4}{441} a^{9} - \frac{1}{441} a^{8} - \frac{19}{441} a^{7} - \frac{25}{63} a^{6} + \frac{23}{49} a^{5} - \frac{25}{441} a^{4} + \frac{115}{441} a^{3} + \frac{46}{147} a^{2} + \frac{8}{147} a - \frac{17}{49}$, $\frac{1}{1323} a^{13} - \frac{1}{1323} a^{12} + \frac{2}{441} a^{11} - \frac{4}{1323} a^{10} + \frac{20}{1323} a^{9} + \frac{2}{1323} a^{8} - \frac{40}{189} a^{7} - \frac{29}{441} a^{6} - \frac{613}{1323} a^{5} + \frac{409}{1323} a^{4} - \frac{1}{147} a^{3} - \frac{209}{441} a^{2} + \frac{25}{147} a - \frac{2}{21}$, $\frac{1}{1323} a^{14} - \frac{1}{1323} a^{12} + \frac{8}{1323} a^{11} - \frac{20}{1323} a^{10} - \frac{17}{1323} a^{9} - \frac{20}{1323} a^{8} - \frac{442}{1323} a^{7} - \frac{13}{189} a^{6} - \frac{41}{441} a^{5} - \frac{332}{1323} a^{4} - \frac{148}{441} a^{3} + \frac{94}{441} a^{2} + \frac{58}{147} a + \frac{4}{147}$, $\frac{1}{66604366388932028112533167171501966161} a^{15} + \frac{1680577617267802742136134485683950}{22201455462977342704177722390500655387} a^{14} - \frac{17818080240911799595886097983532511}{66604366388932028112533167171501966161} a^{13} + \frac{19242502430606489692879137472463570}{66604366388932028112533167171501966161} a^{12} + \frac{420696043108856764644609792985373590}{66604366388932028112533167171501966161} a^{11} + \frac{944713862201052656707684314212297272}{66604366388932028112533167171501966161} a^{10} - \frac{1540848337198558931826959612546451014}{66604366388932028112533167171501966161} a^{9} - \frac{146273353180947969112645270993579}{9514909484133146873219023881643138023} a^{8} + \frac{18983696321485644589463755688198559260}{66604366388932028112533167171501966161} a^{7} - \frac{4340950236628488100818420025263055784}{22201455462977342704177722390500655387} a^{6} - \frac{789369776834696870175925551311170987}{3917903905231295771325480421853056833} a^{5} + \frac{13851134617641243073506744590482577}{74752375296220009104975496264311971} a^{4} - \frac{10787803321807280282754504820326853547}{22201455462977342704177722390500655387} a^{3} + \frac{111657571218819665727957667967056034}{672771377665980081944779466378807739} a^{2} + \frac{1498665136837196982224095028583884608}{7400485154325780901392574130166885129} a + \frac{569729316072159661331015987174236202}{2466828384775260300464191376722295043}$

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:  \( 164472664443 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1638:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1638 are not computed
Character table for t16n1638 is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), 4.4.132741.1, 8.8.5303672097381.1

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 $16$ R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$7$7.8.7.1$x^{8} + 14$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$
7.8.7.1$x^{8} + 14$$8$$1$$7$$D_{8}$$[\ ]_{8}^{2}$
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
6551Data not computed