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Properties

Label 16.8.19719055677...3977.1
Degree $16$
Signature $[8, 4]$
Discriminant $19^{12}\cdot 73^{15}$
Root discriminant $508.08$
Ramified primes $19, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-823982021, 9713737864, -15509537492, 10814479334, -3456857577, -2359973708, 814005396, -12983686, -10053370, 9220860, -2198790, 193350, -32134, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 32134*x^12 + 193350*x^11 - 2198790*x^10 + 9220860*x^9 - 10053370*x^8 - 12983686*x^7 + 814005396*x^6 - 2359973708*x^5 - 3456857577*x^4 + 10814479334*x^3 - 15509537492*x^2 + 9713737864*x - 823982021)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 32134*x^12 + 193350*x^11 - 2198790*x^10 + 9220860*x^9 - 10053370*x^8 - 12983686*x^7 + 814005396*x^6 - 2359973708*x^5 - 3456857577*x^4 + 10814479334*x^3 - 15509537492*x^2 + 9713737864*x - 823982021, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 32134 x^{12} + 193350 x^{11} - 2198790 x^{10} + 9220860 x^{9} - 10053370 x^{8} - 12983686 x^{7} + 814005396 x^{6} - 2359973708 x^{5} - 3456857577 x^{4} + 10814479334 x^{3} - 15509537492 x^{2} + 9713737864 x - 823982021 \)

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:  \(19719055677142144882125300632796548005703977=19^{12}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $508.08$
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:  $19, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{38} a^{4} - \frac{1}{19} a^{3} - \frac{4}{19} a^{2} + \frac{9}{38} a - \frac{13}{38}$, $\frac{1}{38} a^{5} - \frac{6}{19} a^{3} - \frac{7}{38} a^{2} + \frac{5}{38} a + \frac{6}{19}$, $\frac{1}{38} a^{6} + \frac{7}{38} a^{3} - \frac{15}{38} a^{2} + \frac{3}{19} a - \frac{2}{19}$, $\frac{1}{38} a^{7} - \frac{1}{38} a^{3} - \frac{7}{19} a^{2} + \frac{9}{38} a + \frac{15}{38}$, $\frac{1}{2888} a^{8} - \frac{1}{722} a^{7} + \frac{13}{1444} a^{6} + \frac{3}{722} a^{5} - \frac{9}{722} a^{4} - \frac{369}{1444} a^{3} - \frac{1155}{2888} a^{2} + \frac{453}{1444} a - \frac{1389}{2888}$, $\frac{1}{2888} a^{9} + \frac{5}{1444} a^{7} - \frac{9}{722} a^{6} + \frac{3}{722} a^{5} + \frac{15}{1444} a^{4} - \frac{1219}{2888} a^{3} - \frac{33}{1444} a^{2} + \frac{867}{2888} a + \frac{131}{722}$, $\frac{1}{54872} a^{10} - \frac{5}{54872} a^{9} - \frac{3}{54872} a^{8} + \frac{21}{27436} a^{7} - \frac{73}{27436} a^{6} + \frac{135}{27436} a^{5} - \frac{293}{54872} a^{4} + \frac{195}{54872} a^{3} + \frac{6827}{13718} a^{2} - \frac{27369}{54872} a + \frac{17033}{54872}$, $\frac{1}{54872} a^{11} - \frac{9}{54872} a^{9} + \frac{1}{6859} a^{8} + \frac{165}{27436} a^{7} - \frac{97}{27436} a^{6} - \frac{387}{54872} a^{5} - \frac{2}{6859} a^{4} - \frac{2073}{13718} a^{3} + \frac{4283}{27436} a^{2} - \frac{9365}{54872} a - \frac{1417}{6859}$, $\frac{1}{438976} a^{12} + \frac{1}{219488} a^{11} - \frac{17}{438976} a^{9} - \frac{61}{438976} a^{8} + \frac{343}{54872} a^{7} - \frac{341}{438976} a^{6} + \frac{1189}{109744} a^{5} - \frac{1917}{438976} a^{4} - \frac{195237}{438976} a^{3} + \frac{216299}{438976} a^{2} + \frac{187707}{438976} a + \frac{6823}{23104}$, $\frac{1}{17998016} a^{13} + \frac{7}{8999008} a^{12} + \frac{9}{1124876} a^{11} + \frac{79}{17998016} a^{10} + \frac{607}{17998016} a^{9} + \frac{139}{4499504} a^{8} + \frac{224571}{17998016} a^{7} + \frac{947}{118408} a^{6} - \frac{158101}{17998016} a^{5} - \frac{29409}{17998016} a^{4} + \frac{7332239}{17998016} a^{3} + \frac{3520159}{17998016} a^{2} - \frac{3394959}{17998016} a - \frac{1384389}{4499504}$, $\frac{1}{44502320958582495324352} a^{14} - \frac{7}{44502320958582495324352} a^{13} + \frac{6008173322471921}{11125580239645623831088} a^{12} - \frac{144196159739326013}{44502320958582495324352} a^{11} - \frac{56538673404570555}{11125580239645623831088} a^{10} + \frac{2452571599035232719}{44502320958582495324352} a^{9} - \frac{3008956937941732435}{44502320958582495324352} a^{8} - \frac{4265759599577052431}{44502320958582495324352} a^{7} - \frac{99402790852963865695}{44502320958582495324352} a^{6} + \frac{20314093101905740241}{2781395059911405957772} a^{5} - \frac{113903713415014053415}{22251160479291247662176} a^{4} - \frac{48384618666786756147}{22251160479291247662176} a^{3} - \frac{1004245485415735538783}{11125580239645623831088} a^{2} + \frac{4121104370147567068797}{44502320958582495324352} a - \frac{10376462216702750662589}{22251160479291247662176}$, $\frac{1}{3159664788059357168028992} a^{15} + \frac{7}{789916197014839292007248} a^{14} + \frac{83375662148564967}{3159664788059357168028992} a^{13} + \frac{311218807825338995}{3159664788059357168028992} a^{12} + \frac{2461346656752945141}{3159664788059357168028992} a^{11} + \frac{13823622201465551619}{3159664788059357168028992} a^{10} + \frac{79904385066402973067}{1579832394029678584014496} a^{9} - \frac{594044796084183331}{789916197014839292007248} a^{8} + \frac{2082572760043758592811}{789916197014839292007248} a^{7} + \frac{34237210481508353027407}{3159664788059357168028992} a^{6} + \frac{1064619604021736759285}{1579832394029678584014496} a^{5} - \frac{1243870317451477494579}{789916197014839292007248} a^{4} + \frac{381359177744618049720567}{1579832394029678584014496} a^{3} + \frac{1497794920409933751287885}{3159664788059357168028992} a^{2} + \frac{110070528405134176968393}{3159664788059357168028992} a + \frac{666272300372490310672443}{1579832394029678584014496}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 4792824524050000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.1439708022407240137.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$

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
$19$19.8.6.3$x^{8} - 19 x^{4} + 722$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
19.8.6.3$x^{8} - 19 x^{4} + 722$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
73Data not computed