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Properties

Label 16.0.70367540303...0128.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{45}\cdot 7^{4}\cdot 97^{6}$
Root discriminant $63.53$
Ramified primes $2, 7, 97$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group 16T1433

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2485852, 6548432, 20419528, 6066512, 467446, -733488, -125676, 159744, 60652, 5424, -2796, -680, 51, -8, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 - 8*x^13 + 51*x^12 - 680*x^11 - 2796*x^10 + 5424*x^9 + 60652*x^8 + 159744*x^7 - 125676*x^6 - 733488*x^5 + 467446*x^4 + 6066512*x^3 + 20419528*x^2 + 6548432*x + 2485852)
 
gp: K = bnfinit(x^16 - 18*x^14 - 8*x^13 + 51*x^12 - 680*x^11 - 2796*x^10 + 5424*x^9 + 60652*x^8 + 159744*x^7 - 125676*x^6 - 733488*x^5 + 467446*x^4 + 6066512*x^3 + 20419528*x^2 + 6548432*x + 2485852, 1)
 

Normalized defining polynomial

\( x^{16} - 18 x^{14} - 8 x^{13} + 51 x^{12} - 680 x^{11} - 2796 x^{10} + 5424 x^{9} + 60652 x^{8} + 159744 x^{7} - 125676 x^{6} - 733488 x^{5} + 467446 x^{4} + 6066512 x^{3} + 20419528 x^{2} + 6548432 x + 2485852 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(70367540303366615290376880128=2^{45}\cdot 7^{4}\cdot 97^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.53$
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:  $2, 7, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{56290878954820882466609267344728590202172418852270272} a^{15} - \frac{822141987645609975001929974223757293519699635303307}{56290878954820882466609267344728590202172418852270272} a^{14} - \frac{11030804853251230800626265946151396646854962070679065}{56290878954820882466609267344728590202172418852270272} a^{13} + \frac{2287016231337021542917907296332296354132032270074155}{56290878954820882466609267344728590202172418852270272} a^{12} + \frac{1398498016822714246940711844147692175852345383039805}{28145439477410441233304633672364295101086209426135136} a^{11} - \frac{4857081540970882436532684232186600949165888708839667}{28145439477410441233304633672364295101086209426135136} a^{10} + \frac{2099523139444166598215824523569330041103159056231099}{28145439477410441233304633672364295101086209426135136} a^{9} + \frac{11793714794901297559818531355714877669927610383245247}{28145439477410441233304633672364295101086209426135136} a^{8} - \frac{7209873310528738220101955862623573012395696686800463}{28145439477410441233304633672364295101086209426135136} a^{7} - \frac{5329386091840915499684238044913744999995384789762811}{28145439477410441233304633672364295101086209426135136} a^{6} + \frac{3522359203187282025730283391203869448385926904413075}{28145439477410441233304633672364295101086209426135136} a^{5} - \frac{3034463115784942889186524858261999572748952726747625}{28145439477410441233304633672364295101086209426135136} a^{4} - \frac{25253563431221761939323126096395048398886021847685}{129107520538580005657360704919102271105900043239152} a^{3} + \frac{1909420168845092261387478058630719901699152177295391}{14072719738705220616652316836182147550543104713067568} a^{2} - \frac{5837894390441126535278582971765429850962741245525491}{14072719738705220616652316836182147550543104713067568} a - \frac{563520373635547067483500788162884115533280290748139}{14072719738705220616652316836182147550543104713067568}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  $7$
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:  \( 84571085.4576 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1433:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 68 conjugacy class representatives for t16n1433 are not computed
Character table for t16n1433 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.6208.2, 8.0.1233256448.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ $16$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.8.28.20$x^{8} + 8 x^{7} + 8 x^{5} + 30$$8$$1$$28$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$97$97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$