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Properties

Label 16.0.12809387541...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{10}\cdot 13^{6}\cdot 101^{6}$
Root discriminant $57.11$
Ramified primes $2, 5, 13, 101$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T1189

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1069331, 665598, 667787, 318279, 403846, 184903, 98153, 44962, 22085, 8715, 2033, 514, 138, -21, 31, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 31*x^14 - 21*x^13 + 138*x^12 + 514*x^11 + 2033*x^10 + 8715*x^9 + 22085*x^8 + 44962*x^7 + 98153*x^6 + 184903*x^5 + 403846*x^4 + 318279*x^3 + 667787*x^2 + 665598*x + 1069331)
 
gp: K = bnfinit(x^16 - 6*x^15 + 31*x^14 - 21*x^13 + 138*x^12 + 514*x^11 + 2033*x^10 + 8715*x^9 + 22085*x^8 + 44962*x^7 + 98153*x^6 + 184903*x^5 + 403846*x^4 + 318279*x^3 + 667787*x^2 + 665598*x + 1069331, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 31 x^{14} - 21 x^{13} + 138 x^{12} + 514 x^{11} + 2033 x^{10} + 8715 x^{9} + 22085 x^{8} + 44962 x^{7} + 98153 x^{6} + 184903 x^{5} + 403846 x^{4} + 318279 x^{3} + 667787 x^{2} + 665598 x + 1069331 \)

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:  \(12809387541505655522500000000=2^{8}\cdot 5^{10}\cdot 13^{6}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.11$
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:  $2, 5, 13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{221596487050218455857204919271353125922485135} a^{15} - \frac{9359228400287563315472273792806854074785492}{221596487050218455857204919271353125922485135} a^{14} - \frac{20674390471738884409301274632239365232291996}{221596487050218455857204919271353125922485135} a^{13} + \frac{20829421327689112317149045352505697695452232}{221596487050218455857204919271353125922485135} a^{12} - \frac{20021449624939571898774945800060218963505404}{44319297410043691171440983854270625184497027} a^{11} + \frac{63106507085233629704007368192786370910801858}{221596487050218455857204919271353125922485135} a^{10} - \frac{65236252809286764435098995568632933445974097}{221596487050218455857204919271353125922485135} a^{9} + \frac{43529804035714803883239444334481055372129}{44319297410043691171440983854270625184497027} a^{8} + \frac{101912674547396107017246004934935301848941421}{221596487050218455857204919271353125922485135} a^{7} + \frac{1197094161605060285403945867726159697317885}{44319297410043691171440983854270625184497027} a^{6} - \frac{1631615361324269536101493108372868881660337}{44319297410043691171440983854270625184497027} a^{5} + \frac{103910239052715362407481861009266267599475068}{221596487050218455857204919271353125922485135} a^{4} - \frac{19574258174389239805143818352241564207064914}{221596487050218455857204919271353125922485135} a^{3} - \frac{230731656805487407697024512470489394129414}{44319297410043691171440983854270625184497027} a^{2} - \frac{1423123303056582455750464834519993755703167}{221596487050218455857204919271353125922485135} a - \frac{7625042438454621833165219539431650862837004}{44319297410043691171440983854270625184497027}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 2758007.83731 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1189:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1189 are not computed
Character table for t16n1189 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.0.1077480625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
$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}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
13.8.6.4$x^{8} - 13 x^{4} + 338$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$