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Properties

Label 16.8.47347598743...8937.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{12}\cdot 73^{15}$
Root discriminant $127.26$
Ramified primes $3, 73$
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![-4157, 34036, -113712, 184874, -97045, -162276, 339220, -253070, 69102, 3880, -8498, 4134, -598, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 598*x^12 + 4134*x^11 - 8498*x^10 + 3880*x^9 + 69102*x^8 - 253070*x^7 + 339220*x^6 - 162276*x^5 - 97045*x^4 + 184874*x^3 - 113712*x^2 + 34036*x - 4157)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 598*x^12 + 4134*x^11 - 8498*x^10 + 3880*x^9 + 69102*x^8 - 253070*x^7 + 339220*x^6 - 162276*x^5 - 97045*x^4 + 184874*x^3 - 113712*x^2 + 34036*x - 4157, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 598 x^{12} + 4134 x^{11} - 8498 x^{10} + 3880 x^{9} + 69102 x^{8} - 253070 x^{7} + 339220 x^{6} - 162276 x^{5} - 97045 x^{4} + 184874 x^{3} - 113712 x^{2} + 34036 x - 4157 \)

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:  \(4734759874359678578687627415368937=3^{12}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $127.26$
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:  $3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{6} a^{5} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{3} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{36} a^{8} + \frac{1}{18} a^{7} - \frac{1}{18} a^{6} + \frac{1}{18} a^{5} - \frac{1}{18} a^{4} + \frac{2}{9} a^{3} - \frac{17}{36} a^{2} - \frac{1}{9} a - \frac{17}{36}$, $\frac{1}{36} a^{9} + \frac{5}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{4}{9}$, $\frac{1}{108} a^{10} + \frac{1}{108} a^{9} - \frac{1}{18} a^{7} + \frac{1}{18} a^{6} + \frac{1}{36} a^{4} - \frac{1}{4} a^{3} - \frac{1}{12} a^{2} - \frac{35}{108} a - \frac{7}{54}$, $\frac{1}{108} a^{11} - \frac{1}{108} a^{9} + \frac{1}{18} a^{7} - \frac{1}{36} a^{5} - \frac{1}{18} a^{4} + \frac{5}{18} a^{3} + \frac{4}{27} a^{2} - \frac{1}{36} a - \frac{17}{54}$, $\frac{1}{432} a^{12} - \frac{1}{216} a^{11} + \frac{1}{144} a^{9} + \frac{1}{144} a^{8} - \frac{1}{18} a^{7} - \frac{1}{48} a^{6} + \frac{1}{36} a^{5} - \frac{7}{144} a^{4} + \frac{103}{432} a^{3} + \frac{187}{432} a^{2} - \frac{23}{144} a + \frac{71}{144}$, $\frac{1}{432} a^{13} - \frac{1}{432} a^{10} + \frac{1}{432} a^{9} - \frac{1}{72} a^{8} + \frac{5}{144} a^{7} + \frac{1}{24} a^{6} + \frac{5}{144} a^{5} + \frac{1}{432} a^{4} + \frac{71}{144} a^{3} - \frac{5}{144} a^{2} + \frac{155}{432} a - \frac{1}{216}$, $\frac{1}{47551353264} a^{14} - \frac{7}{47551353264} a^{13} + \frac{13533235}{47551353264} a^{12} - \frac{81199319}{47551353264} a^{11} + \frac{41866103}{11887838316} a^{10} - \frac{5812196}{2971959579} a^{9} - \frac{90359759}{7925225544} a^{8} + \frac{611137603}{15850451088} a^{7} - \frac{176098643}{7925225544} a^{6} - \frac{686139155}{11887838316} a^{5} - \frac{3763251109}{47551353264} a^{4} + \frac{13979139031}{47551353264} a^{3} - \frac{12803114987}{47551353264} a^{2} + \frac{2545159051}{23775676632} a - \frac{16742134415}{47551353264}$, $\frac{1}{47551353264} a^{15} + \frac{2255531}{7925225544} a^{13} + \frac{6766663}{23775676632} a^{12} + \frac{13119829}{15850451088} a^{11} + \frac{16556261}{3962612772} a^{10} + \frac{63873209}{23775676632} a^{9} - \frac{71202905}{5283483696} a^{8} - \frac{159045715}{5283483696} a^{7} - \frac{1107737041}{23775676632} a^{6} + \frac{235711123}{5283483696} a^{5} + \frac{290569363}{3962612772} a^{4} - \frac{5025924149}{23775676632} a^{3} - \frac{6896464049}{15850451088} a^{2} - \frac{1481764675}{15850451088} a + \frac{10929538723}{47551353264}$

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:  \( 111926804231 \) (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, 8.8.894839280046857.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$

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
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
73Data not computed