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Properties

Label 16.8.22854104234...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{2}\cdot 101^{6}$
Root discriminant $38.45$
Ramified primes $2, 5, 29, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1429

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![701, -9142, 41797, -80214, 60884, 6880, -39356, 21856, -1700, -2312, 625, 136, -124, 30, 1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + x^14 + 30*x^13 - 124*x^12 + 136*x^11 + 625*x^10 - 2312*x^9 - 1700*x^8 + 21856*x^7 - 39356*x^6 + 6880*x^5 + 60884*x^4 - 80214*x^3 + 41797*x^2 - 9142*x + 701)
 
gp: K = bnfinit(x^16 - 4*x^15 + x^14 + 30*x^13 - 124*x^12 + 136*x^11 + 625*x^10 - 2312*x^9 - 1700*x^8 + 21856*x^7 - 39356*x^6 + 6880*x^5 + 60884*x^4 - 80214*x^3 + 41797*x^2 - 9142*x + 701, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + x^{14} + 30 x^{13} - 124 x^{12} + 136 x^{11} + 625 x^{10} - 2312 x^{9} - 1700 x^{8} + 21856 x^{7} - 39356 x^{6} + 6880 x^{5} + 60884 x^{4} - 80214 x^{3} + 41797 x^{2} - 9142 x + 701 \)

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:  \(22854104234379289600000000=2^{16}\cdot 5^{8}\cdot 29^{2}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.45$
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:  $2, 5, 29, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{41} a^{14} - \frac{6}{41} a^{13} + \frac{14}{41} a^{12} - \frac{4}{41} a^{11} - \frac{20}{41} a^{10} + \frac{8}{41} a^{9} + \frac{15}{41} a^{8} + \frac{3}{41} a^{7} - \frac{10}{41} a^{6} - \frac{15}{41} a^{5} - \frac{17}{41} a^{4} + \frac{11}{41} a^{3} + \frac{1}{41} a^{2} - \frac{9}{41} a - \frac{4}{41}$, $\frac{1}{638374046399889893905953551843} a^{15} - \frac{5164387488963412591811575510}{638374046399889893905953551843} a^{14} + \frac{65576144696986557613672846}{590540283441156238580900603} a^{13} - \frac{313701698274661454033431493010}{638374046399889893905953551843} a^{12} - \frac{265889687709981756653015571129}{638374046399889893905953551843} a^{11} - \frac{36790140103336132169233901994}{638374046399889893905953551843} a^{10} - \frac{82742363120311624671014526027}{638374046399889893905953551843} a^{9} - \frac{215915695747288938991104393992}{638374046399889893905953551843} a^{8} + \frac{28607797345084851185983896696}{638374046399889893905953551843} a^{7} - \frac{224377484885312680781755088277}{638374046399889893905953551843} a^{6} + \frac{176527235351656695149190214709}{638374046399889893905953551843} a^{5} - \frac{3214381886044791544862547148}{15570098692680241314779354923} a^{4} - \frac{66741128901949825093285490692}{638374046399889893905953551843} a^{3} - \frac{259649566263701829434060137105}{638374046399889893905953551843} a^{2} + \frac{112697843501534226740966307620}{638374046399889893905953551843} a + \frac{218344076891786922379830259205}{638374046399889893905953551843}$

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:  \( 5512813.02261 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1429:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 77 conjugacy class representatives for t16n1429 are not computed
Character table for t16n1429 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.164848160000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$101$101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
101.8.6.1$x^{8} - 707 x^{4} + 826281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$