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Properties

Label 16.8.16623228109...8125.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 29^{9}\cdot 41^{2}\cdot 1321^{2}$
Root discriminant $58.05$
Ramified primes $5, 29, 41, 1321$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1702

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![499, 4577, 5638, -15340, -25447, -382, 5387, -2917, 289, 250, -526, 36, -52, 4, 6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 6*x^14 + 4*x^13 - 52*x^12 + 36*x^11 - 526*x^10 + 250*x^9 + 289*x^8 - 2917*x^7 + 5387*x^6 - 382*x^5 - 25447*x^4 - 15340*x^3 + 5638*x^2 + 4577*x + 499)
 
gp: K = bnfinit(x^16 - 2*x^15 + 6*x^14 + 4*x^13 - 52*x^12 + 36*x^11 - 526*x^10 + 250*x^9 + 289*x^8 - 2917*x^7 + 5387*x^6 - 382*x^5 - 25447*x^4 - 15340*x^3 + 5638*x^2 + 4577*x + 499, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 6 x^{14} + 4 x^{13} - 52 x^{12} + 36 x^{11} - 526 x^{10} + 250 x^{9} + 289 x^{8} - 2917 x^{7} + 5387 x^{6} - 382 x^{5} - 25447 x^{4} - 15340 x^{3} + 5638 x^{2} + 4577 x + 499 \)

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:  \(16623228109216115106386328125=5^{8}\cdot 29^{9}\cdot 41^{2}\cdot 1321^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.05$
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:  $5, 29, 41, 1321$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{605937268095824607187428131656127} a^{15} + \frac{67515545076263010113079348900838}{605937268095824607187428131656127} a^{14} - \frac{245848589811648076267453924236978}{605937268095824607187428131656127} a^{13} + \frac{221153985908701708410930961711050}{605937268095824607187428131656127} a^{12} + \frac{201609338836218235795364996058883}{605937268095824607187428131656127} a^{11} + \frac{15407539949863763627222670160939}{86562466870832086741061161665161} a^{10} + \frac{160997349354823972671051778231439}{605937268095824607187428131656127} a^{9} + \frac{165777698356036348583825930003846}{605937268095824607187428131656127} a^{8} - \frac{120220621361241916618963361057963}{605937268095824607187428131656127} a^{7} - \frac{47837357198927747790463440120237}{605937268095824607187428131656127} a^{6} + \frac{212911118460425952205334459823638}{605937268095824607187428131656127} a^{5} + \frac{247100550009755510145623697118843}{605937268095824607187428131656127} a^{4} + \frac{175187838921025131869492645954832}{605937268095824607187428131656127} a^{3} - \frac{190492083908131077181040200624940}{605937268095824607187428131656127} a^{2} - \frac{96695828793947302905820908869839}{605937268095824607187428131656127} a - \frac{41444686938901969428663054266962}{605937268095824607187428131656127}$

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:  \( 76558798.2461 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1702:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 104 conjugacy class representatives for t16n1702 are not computed
Character table for t16n1702 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.28468375625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.4.2$x^{8} - 24389 x^{2} + 13438339$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
1321Data not computed