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Properties

Label 16.8.13657651539...2656.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 3^{8}\cdot 7^{6}\cdot 1567^{2}$
Root discriminant $101.97$
Ramified primes $2, 3, 7, 1567$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14596103, 63626096, -109532856, 92061888, -33384068, -2574120, 5984592, -1488608, -42220, 40480, -2240, 4352, -836, -120, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 120*x^13 - 836*x^12 + 4352*x^11 - 2240*x^10 + 40480*x^9 - 42220*x^8 - 1488608*x^7 + 5984592*x^6 - 2574120*x^5 - 33384068*x^4 + 92061888*x^3 - 109532856*x^2 + 63626096*x - 14596103)
 
gp: K = bnfinit(x^16 - 120*x^13 - 836*x^12 + 4352*x^11 - 2240*x^10 + 40480*x^9 - 42220*x^8 - 1488608*x^7 + 5984592*x^6 - 2574120*x^5 - 33384068*x^4 + 92061888*x^3 - 109532856*x^2 + 63626096*x - 14596103, 1)
 

Normalized defining polynomial

\( x^{16} - 120 x^{13} - 836 x^{12} + 4352 x^{11} - 2240 x^{10} + 40480 x^{9} - 42220 x^{8} - 1488608 x^{7} + 5984592 x^{6} - 2574120 x^{5} - 33384068 x^{4} + 92061888 x^{3} - 109532856 x^{2} + 63626096 x - 14596103 \)

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:  \(136576515395793105198700252102656=2^{56}\cdot 3^{8}\cdot 7^{6}\cdot 1567^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.97$
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:  $2, 3, 7, 1567$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{16340872867427513408666712843240196228718006821707577975} a^{15} + \frac{2728805147178320723751436949009862912963318548515456663}{16340872867427513408666712843240196228718006821707577975} a^{14} + \frac{4692900873427742759581942731012143596499328548850032219}{16340872867427513408666712843240196228718006821707577975} a^{13} - \frac{76585372054579429594749169973757605756347442439057408}{527124931207339142215055253007748265442516349087341225} a^{12} - \frac{1508536174272091710823123236263540312773846482636118677}{3268174573485502681733342568648039245743601364341515595} a^{11} - \frac{4548882749699377052416266984417923593349291074177087753}{16340872867427513408666712843240196228718006821707577975} a^{10} + \frac{4287129895492513349798459289963911187503855142061348621}{16340872867427513408666712843240196228718006821707577975} a^{9} + \frac{3787751085581213460650884699875115386174005913652650803}{16340872867427513408666712843240196228718006821707577975} a^{8} + \frac{59638680271357977706614330729004134738747555761431357}{961227815731030200509806637837658601689294518923975175} a^{7} - \frac{5341445110995871950827446663722343486885306090808790811}{16340872867427513408666712843240196228718006821707577975} a^{6} - \frac{7741049668296554560681066658305806600005342734388118501}{16340872867427513408666712843240196228718006821707577975} a^{5} + \frac{2834199108193227461067056431458860350137930453400656817}{16340872867427513408666712843240196228718006821707577975} a^{4} + \frac{4798310003123777369561543017837047295994601300220104928}{16340872867427513408666712843240196228718006821707577975} a^{3} + \frac{2209332018235087031835802024969227535918189897195312602}{16340872867427513408666712843240196228718006821707577975} a^{2} + \frac{613213127430378667429365438520449321766468549624530444}{3268174573485502681733342568648039245743601364341515595} a + \frac{5316687445563452260767800799328308533914955090912866931}{16340872867427513408666712843240196228718006821707577975}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 11124092813.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.4.129024.1, 4.4.14336.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.66588770304.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
1567Data not computed