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Properties

Label 16.8.57135260585...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{10}\cdot 29^{2}\cdot 101^{6}$
Root discriminant $47.02$
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![3239, -2446, -3858, 3166, -4216, 3546, 3265, -4612, 1636, -82, -176, 158, -140, 54, 9, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 9*x^14 + 54*x^13 - 140*x^12 + 158*x^11 - 176*x^10 - 82*x^9 + 1636*x^8 - 4612*x^7 + 3265*x^6 + 3546*x^5 - 4216*x^4 + 3166*x^3 - 3858*x^2 - 2446*x + 3239)
 
gp: K = bnfinit(x^16 - 8*x^15 + 9*x^14 + 54*x^13 - 140*x^12 + 158*x^11 - 176*x^10 - 82*x^9 + 1636*x^8 - 4612*x^7 + 3265*x^6 + 3546*x^5 - 4216*x^4 + 3166*x^3 - 3858*x^2 - 2446*x + 3239, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 9 x^{14} + 54 x^{13} - 140 x^{12} + 158 x^{11} - 176 x^{10} - 82 x^{9} + 1636 x^{8} - 4612 x^{7} + 3265 x^{6} + 3546 x^{5} - 4216 x^{4} + 3166 x^{3} - 3858 x^{2} - 2446 x + 3239 \)

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:  \(571352605859482240000000000=2^{16}\cdot 5^{10}\cdot 29^{2}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.02$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{3}{8} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{3}{16} a^{10} - \frac{1}{16} a^{9} + \frac{5}{16} a^{8} - \frac{3}{16} a^{7} + \frac{1}{16} a^{6} - \frac{3}{16} a^{5} + \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{3}{16} a + \frac{5}{16}$, $\frac{1}{32} a^{14} - \frac{1}{16} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{3}{8} a^{8} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{5}{16} a^{5} + \frac{9}{32} a^{4} + \frac{1}{8} a^{3} + \frac{7}{32} a^{2} + \frac{1}{16} a - \frac{1}{32}$, $\frac{1}{4205915542547442203290016} a^{15} - \frac{21375378730731836092235}{4205915542547442203290016} a^{14} + \frac{54817628821919010781691}{2102957771273721101645008} a^{13} - \frac{5654319410637448994025}{1051478885636860550822504} a^{12} - \frac{29837113269669680492487}{1051478885636860550822504} a^{11} - \frac{22174935068020831503235}{2102957771273721101645008} a^{10} + \frac{596759786018794405598451}{2102957771273721101645008} a^{9} - \frac{348176636877101016167941}{1051478885636860550822504} a^{8} + \frac{52106657337921648481541}{1051478885636860550822504} a^{7} + \frac{654777716368638790505}{5592972795940747610758} a^{6} + \frac{1631019780064577738346725}{4205915542547442203290016} a^{5} - \frac{1895941271448970249930485}{4205915542547442203290016} a^{4} + \frac{522757780707217628227079}{4205915542547442203290016} a^{3} + \frac{369972997543187112658249}{4205915542547442203290016} a^{2} + \frac{255134591435082696542703}{4205915542547442203290016} a + \frac{23251615300289020522813}{102583305915791273250976}$

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:  \( 31165988.3475 \) (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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ 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
$2$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
101Data not computed