↑

Properties

Label 16.8.11748739546...3136.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{28}\cdot 17^{8}\cdot 89^{4}$
Root discriminant $42.60$
Ramified primes $2, 17, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3$ (as 16T373)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -14, -145, 1094, 7544, 7110, -3455, -2022, 558, -474, 145, 66, -40, 2, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - x^14 + 2*x^13 - 40*x^12 + 66*x^11 + 145*x^10 - 474*x^9 + 558*x^8 - 2022*x^7 - 3455*x^6 + 7110*x^5 + 7544*x^4 + 1094*x^3 - 145*x^2 - 14*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - x^14 + 2*x^13 - 40*x^12 + 66*x^11 + 145*x^10 - 474*x^9 + 558*x^8 - 2022*x^7 - 3455*x^6 + 7110*x^5 + 7544*x^4 + 1094*x^3 - 145*x^2 - 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - x^{14} + 2 x^{13} - 40 x^{12} + 66 x^{11} + 145 x^{10} - 474 x^{9} + 558 x^{8} - 2022 x^{7} - 3455 x^{6} + 7110 x^{5} + 7544 x^{4} + 1094 x^{3} - 145 x^{2} - 14 x + 1 \)

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:  \(117487395465924089674203136=2^{28}\cdot 17^{8}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.60$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{48} a^{12} - \frac{1}{24} a^{11} - \frac{1}{48} a^{10} - \frac{5}{24} a^{9} - \frac{1}{16} a^{8} + \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{3} a^{5} + \frac{5}{16} a^{4} - \frac{1}{24} a^{3} - \frac{11}{48} a^{2} + \frac{1}{24} a - \frac{13}{48}$, $\frac{1}{96} a^{13} - \frac{1}{96} a^{12} - \frac{1}{32} a^{11} - \frac{11}{96} a^{10} + \frac{11}{96} a^{9} - \frac{23}{96} a^{8} - \frac{1}{12} a^{7} - \frac{1}{24} a^{6} + \frac{47}{96} a^{5} + \frac{37}{96} a^{4} + \frac{35}{96} a^{3} - \frac{11}{32} a^{2} + \frac{13}{96} a + \frac{35}{96}$, $\frac{1}{576} a^{14} - \frac{1}{288} a^{13} - \frac{1}{96} a^{12} - \frac{35}{288} a^{10} - \frac{7}{96} a^{9} + \frac{41}{192} a^{8} + \frac{1}{16} a^{7} - \frac{47}{192} a^{6} - \frac{5}{32} a^{5} + \frac{89}{288} a^{4} + \frac{7}{48} a^{3} + \frac{13}{32} a^{2} + \frac{55}{288} a + \frac{161}{576}$, $\frac{1}{224027353151756928} a^{15} - \frac{41803988531383}{74675784383918976} a^{14} + \frac{10963041786349}{7000854785992404} a^{13} + \frac{322306812115847}{37337892191959488} a^{12} + \frac{696290147268301}{112013676575878464} a^{11} + \frac{80101673025230}{1750213696498101} a^{10} + \frac{5081213599762785}{24891928127972992} a^{9} + \frac{427079834927435}{1736646148463232} a^{8} + \frac{4628264510973533}{74675784383918976} a^{7} + \frac{15668288097835327}{74675784383918976} a^{6} + \frac{3131234383551703}{14001709571984808} a^{5} + \frac{26139114797839195}{112013676575878464} a^{4} - \frac{17409100938205915}{37337892191959488} a^{3} + \frac{13353793501239277}{28003419143969616} a^{2} - \frac{3319391844052161}{24891928127972992} a + \frac{33052012728828181}{224027353151756928}$

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:  \( 87688996.0868 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T373):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), 4.4.205768.2, \(\Q(\sqrt{2}, \sqrt{17})\), 4.4.205768.1, 8.8.2709790068736.5, 8.4.121788317696.2, 8.4.121788317696.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$