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Properties

Label 16.4.26703352420...3952.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{54}\cdot 7^{8}\cdot 137^{3}$
Root discriminant $69.05$
Ramified primes $2, 7, 137$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1461

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-192398, 739504, -1192448, 1586144, -1205276, 816464, -282080, 83464, -12978, 1168, -2568, -296, 240, -112, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 240*x^12 - 296*x^11 - 2568*x^10 + 1168*x^9 - 12978*x^8 + 83464*x^7 - 282080*x^6 + 816464*x^5 - 1205276*x^4 + 1586144*x^3 - 1192448*x^2 + 739504*x - 192398)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 240*x^12 - 296*x^11 - 2568*x^10 + 1168*x^9 - 12978*x^8 + 83464*x^7 - 282080*x^6 + 816464*x^5 - 1205276*x^4 + 1586144*x^3 - 1192448*x^2 + 739504*x - 192398, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 112 x^{13} + 240 x^{12} - 296 x^{11} - 2568 x^{10} + 1168 x^{9} - 12978 x^{8} + 83464 x^{7} - 282080 x^{6} + 816464 x^{5} - 1205276 x^{4} + 1586144 x^{3} - 1192448 x^{2} + 739504 x - 192398 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(267033524201279981591516413952=2^{54}\cdot 7^{8}\cdot 137^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.05$
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:  $2, 7, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1115} a^{14} + \frac{526}{1115} a^{13} - \frac{107}{1115} a^{12} + \frac{538}{1115} a^{11} + \frac{431}{1115} a^{10} + \frac{527}{1115} a^{9} - \frac{87}{1115} a^{8} + \frac{186}{1115} a^{7} + \frac{24}{223} a^{6} + \frac{542}{1115} a^{5} - \frac{522}{1115} a^{4} + \frac{457}{1115} a^{3} - \frac{104}{1115} a^{2} + \frac{534}{1115} a - \frac{369}{1115}$, $\frac{1}{901336051536417538892826299228094783940070595} a^{15} - \frac{1682865112024342884038124050415442317832}{33382816723571019958993566638077584590372985} a^{14} + \frac{46741153982261247035396005676153838236363617}{100148450170713059876980699914232753771118955} a^{13} - \frac{292550290055680726724888153887173771766665142}{901336051536417538892826299228094783940070595} a^{12} - \frac{97501086190234313072582249727140291937714104}{901336051536417538892826299228094783940070595} a^{11} + \frac{18893385005158473262078172754439151401970818}{100148450170713059876980699914232753771118955} a^{10} - \frac{23024265157289571844933945304564501435720224}{300445350512139179630942099742698261313356865} a^{9} - \frac{2940158036396772468488380807101269201800454}{901336051536417538892826299228094783940070595} a^{8} - \frac{72848867242437421788177427247655200430652865}{180267210307283507778565259845618956788014119} a^{7} - \frac{111524568998492744150147763041776910727244258}{901336051536417538892826299228094783940070595} a^{6} - \frac{339672469526311212104162530994923659233132287}{901336051536417538892826299228094783940070595} a^{5} - \frac{106537245319845052448175919169277957138470276}{300445350512139179630942099742698261313356865} a^{4} - \frac{211128977420609328819466078845273100257628034}{901336051536417538892826299228094783940070595} a^{3} + \frac{431211328476414384755682640918621390272186799}{901336051536417538892826299228094783940070595} a^{2} + \frac{15022875182521840183712519125196503247206592}{300445350512139179630942099742698261313356865} a - \frac{61356909665373102527318923848701770500821380}{180267210307283507778565259845618956788014119}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 103699095.403 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1461:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1461 are not computed
Character table for t16n1461 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{14}) \), 4.4.7168.1 x2, 4.4.25088.1 x2, \(\Q(\sqrt{2}, \sqrt{7})\), 8.8.10070523904.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
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}$
137Data not computed