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Properties

Label 16.8.13854815685...5057.3
Degree $16$
Signature $[8, 4]$
Discriminant $41^{15}\cdot 73^{15}$
Root discriminant $1814.89$
Ramified primes $41, 73$
Class number $48$ (GRH)
Class group $[48]$ (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![10792143313968857, 0, -1507021151328107, 0, -11542208172049, 0, 450184977646, 0, 4061186735, 0, -8613854, 0, -125706, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 125706*x^12 - 8613854*x^10 + 4061186735*x^8 + 450184977646*x^6 - 11542208172049*x^4 - 1507021151328107*x^2 + 10792143313968857)
 
gp: K = bnfinit(x^16 - 125706*x^12 - 8613854*x^10 + 4061186735*x^8 + 450184977646*x^6 - 11542208172049*x^4 - 1507021151328107*x^2 + 10792143313968857, 1)
 

Normalized defining polynomial

\( x^{16} - 125706 x^{12} - 8613854 x^{10} + 4061186735 x^{8} + 450184977646 x^{6} - 11542208172049 x^{4} - 1507021151328107 x^{2} + 10792143313968857 \)

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:  \(13854815685101981032101054249860279864869741101695057=41^{15}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1814.89$
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:  $41, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{22} a^{10} + \frac{1}{22} a^{8} - \frac{9}{22} a^{6} + \frac{1}{11} a^{4} - \frac{1}{2} a^{3} - \frac{1}{11} a^{2} - \frac{1}{2} a + \frac{9}{22}$, $\frac{1}{22} a^{11} + \frac{1}{22} a^{9} - \frac{9}{22} a^{7} + \frac{1}{11} a^{5} - \frac{1}{2} a^{4} - \frac{1}{11} a^{3} - \frac{1}{2} a^{2} + \frac{9}{22} a$, $\frac{1}{8294} a^{12} + \frac{69}{8294} a^{10} + \frac{961}{8294} a^{8} + \frac{410}{4147} a^{6} - \frac{1}{2} a^{5} - \frac{978}{4147} a^{4} - \frac{1}{2} a^{3} - \frac{83}{8294} a^{2} + \frac{471}{4147}$, $\frac{1}{8294} a^{13} + \frac{69}{8294} a^{11} + \frac{961}{8294} a^{9} + \frac{410}{4147} a^{7} - \frac{1}{2} a^{6} - \frac{978}{4147} a^{5} - \frac{1}{2} a^{4} - \frac{83}{8294} a^{3} + \frac{471}{4147} a$, $\frac{1}{73475094534684388225197627270300345359344906} a^{14} + \frac{366929774094222915417203835499726550099}{73475094534684388225197627270300345359344906} a^{12} + \frac{138115108429539140738000772831649719012971}{36737547267342194112598813635150172679672453} a^{10} + \frac{11481017019256205394936158940846769044959623}{73475094534684388225197627270300345359344906} a^{8} - \frac{1}{2} a^{7} - \frac{760385714186237601426160966523031190133607}{6679554048607671656836147933663667759940446} a^{6} - \frac{1}{2} a^{5} + \frac{17247811117309325588946281170925923672528245}{73475094534684388225197627270300345359344906} a^{4} - \frac{1}{2} a^{3} + \frac{11822482346018067312452642271089884856624362}{36737547267342194112598813635150172679672453} a^{2} - \frac{1}{2} a + \frac{10051827196595976552710093158665167162469955}{73475094534684388225197627270300345359344906}$, $\frac{1}{139521342686250442010110198040182433700442526589058} a^{15} + \frac{1092727055655514499779155229651608407911}{36737547267342194112598813635150172679672453} a^{13} + \frac{105605694823895341729500070484752006982418615227}{69760671343125221005055099020091216850221263294529} a^{11} + \frac{7044774310880064492982081423864275506104721810203}{139521342686250442010110198040182433700442526589058} a^{9} - \frac{1480993442352551346594394869560119478064304917337}{5366205487932709308081161463083939757709327945733} a^{7} + \frac{66009019390990234886965858293049593762942748592899}{139521342686250442010110198040182433700442526589058} a^{5} + \frac{24550790628597850437126189474759036316971699704465}{139521342686250442010110198040182433700442526589058} a^{3} - \frac{1}{2} a^{2} - \frac{65029104967362924829436021063848838765089905308109}{139521342686250442010110198040182433700442526589058} a - \frac{1}{2}$

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

Class group and class number

$C_{48}$, which has order $48$ (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:  \( 17049668656900000000 \) (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

\(\Q(\sqrt{2993}) \), 4.4.26811440657.3, 8.8.2151528076860770946805457.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
41Data not computed
73Data not computed