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Properties

Label 16.8.23085241309...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{8}\cdot 5^{14}\cdot 7^{8}$
Root discriminant $18.74$
Ramified primes $3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_8:C_2):C_2$ (as 16T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, -23, 1, 46, 2, -100, 38, 89, -81, 15, 29, -39, 13, 8, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 8*x^14 + 13*x^13 - 39*x^12 + 29*x^11 + 15*x^10 - 81*x^9 + 89*x^8 + 38*x^7 - 100*x^6 + 2*x^5 + 46*x^4 + x^3 - 23*x^2 + 7*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 8*x^14 + 13*x^13 - 39*x^12 + 29*x^11 + 15*x^10 - 81*x^9 + 89*x^8 + 38*x^7 - 100*x^6 + 2*x^5 + 46*x^4 + x^3 - 23*x^2 + 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 8 x^{14} + 13 x^{13} - 39 x^{12} + 29 x^{11} + 15 x^{10} - 81 x^{9} + 89 x^{8} + 38 x^{7} - 100 x^{6} + 2 x^{5} + 46 x^{4} + x^{3} - 23 x^{2} + 7 x + 1 \)

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:  \(230852413092041015625=3^{8}\cdot 5^{14}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.74$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{158227439} a^{15} - \frac{64218149}{158227439} a^{14} - \frac{74345525}{158227439} a^{13} + \frac{70907621}{158227439} a^{12} + \frac{44783291}{158227439} a^{11} + \frac{57316642}{158227439} a^{10} + \frac{19140733}{158227439} a^{9} - \frac{63964472}{158227439} a^{8} + \frac{40457307}{158227439} a^{7} - \frac{1987814}{158227439} a^{6} - \frac{60162484}{158227439} a^{5} + \frac{78596422}{158227439} a^{4} - \frac{45103181}{158227439} a^{3} + \frac{16042288}{158227439} a^{2} - \frac{64152205}{158227439} a - \frac{76390122}{158227439}$

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:  \( 17701.471457 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}:C_2$ (as 16T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $(C_8:C_2):C_2$
Character table for $(C_8:C_2):C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{15})^+\), 4.4.6125.1, \(\Q(\sqrt{5}, \sqrt{21})\), 8.8.3038765625.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
3Data not computed
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
7Data not computed