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Properties

Label 16.8.29630251663...0000.4
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $33.84$
Ramified primes $2, 5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-839, 4472, 2748, -12536, 597, 8652, -5222, -2608, 3200, -48, -684, 224, 42, -36, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 - 36*x^13 + 42*x^12 + 224*x^11 - 684*x^10 - 48*x^9 + 3200*x^8 - 2608*x^7 - 5222*x^6 + 8652*x^5 + 597*x^4 - 12536*x^3 + 2748*x^2 + 4472*x - 839)
 
gp: K = bnfinit(x^16 - 2*x^14 - 36*x^13 + 42*x^12 + 224*x^11 - 684*x^10 - 48*x^9 + 3200*x^8 - 2608*x^7 - 5222*x^6 + 8652*x^5 + 597*x^4 - 12536*x^3 + 2748*x^2 + 4472*x - 839, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{14} - 36 x^{13} + 42 x^{12} + 224 x^{11} - 684 x^{10} - 48 x^{9} + 3200 x^{8} - 2608 x^{7} - 5222 x^{6} + 8652 x^{5} + 597 x^{4} - 12536 x^{3} + 2748 x^{2} + 4472 x - 839 \)

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:  \(2963025166336000000000000=2^{32}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.84$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} + \frac{2}{7} a^{12} + \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7}$, $\frac{1}{49} a^{14} - \frac{1}{49} a^{13} + \frac{11}{49} a^{12} - \frac{10}{49} a^{11} - \frac{12}{49} a^{10} + \frac{18}{49} a^{9} - \frac{13}{49} a^{8} - \frac{15}{49} a^{7} + \frac{3}{7} a^{6} - \frac{16}{49} a^{5} - \frac{5}{49} a^{4} - \frac{12}{49} a^{3} + \frac{10}{49} a^{2} + \frac{1}{49} a - \frac{24}{49}$, $\frac{1}{37295048067808603007024887} a^{15} + \frac{154888566793979167610282}{37295048067808603007024887} a^{14} + \frac{49239293806818678882241}{5327864009686943286717841} a^{13} - \frac{12854731887664754786428213}{37295048067808603007024887} a^{12} + \frac{10050313725916535809858796}{37295048067808603007024887} a^{11} - \frac{10136289753537373063756526}{37295048067808603007024887} a^{10} - \frac{5646124337361940542431080}{37295048067808603007024887} a^{9} + \frac{15715690159220099603029954}{37295048067808603007024887} a^{8} - \frac{9611040175419639612159060}{37295048067808603007024887} a^{7} + \frac{16933692134213083612085050}{37295048067808603007024887} a^{6} + \frac{6173264686570922202783872}{37295048067808603007024887} a^{5} + \frac{16170767449074003087830791}{37295048067808603007024887} a^{4} - \frac{18326752383499694822987820}{37295048067808603007024887} a^{3} - \frac{16578108651293372074619358}{37295048067808603007024887} a^{2} + \frac{3087544283041706745319357}{37295048067808603007024887} a - \frac{5713141242716570692570609}{37295048067808603007024887}$

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:  \( 1644018.76455 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed