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Properties

Label 16.0.28596554320...5089.5
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 19^{8}$
Root discriminant $52.00$
Ramified primes $17, 19$
Class number $104$ (GRH)
Class group $[104]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1191497, -1608706, 1309611, -1288536, 1360137, -908136, 320296, -10952, -35903, 9490, 3280, -1806, 18, 140, -18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 18*x^14 + 140*x^13 + 18*x^12 - 1806*x^11 + 3280*x^10 + 9490*x^9 - 35903*x^8 - 10952*x^7 + 320296*x^6 - 908136*x^5 + 1360137*x^4 - 1288536*x^3 + 1309611*x^2 - 1608706*x + 1191497)
 
gp: K = bnfinit(x^16 - 4*x^15 - 18*x^14 + 140*x^13 + 18*x^12 - 1806*x^11 + 3280*x^10 + 9490*x^9 - 35903*x^8 - 10952*x^7 + 320296*x^6 - 908136*x^5 + 1360137*x^4 - 1288536*x^3 + 1309611*x^2 - 1608706*x + 1191497, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 18 x^{14} + 140 x^{13} + 18 x^{12} - 1806 x^{11} + 3280 x^{10} + 9490 x^{9} - 35903 x^{8} - 10952 x^{7} + 320296 x^{6} - 908136 x^{5} + 1360137 x^{4} - 1288536 x^{3} + 1309611 x^{2} - 1608706 x + 1191497 \)

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:  \(2859655432078149808865465089=17^{14}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.00$
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:  $17, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{34} a^{8} - \frac{1}{17} a^{7} + \frac{3}{17} a^{6} + \frac{7}{17} a^{5} - \frac{15}{34} a^{4} - \frac{7}{17} a^{3} - \frac{11}{34} a^{2} - \frac{15}{34} a + \frac{1}{34}$, $\frac{1}{34} a^{9} + \frac{1}{17} a^{7} - \frac{4}{17} a^{6} + \frac{13}{34} a^{5} - \frac{5}{17} a^{4} - \frac{5}{34} a^{3} - \frac{3}{34} a^{2} + \frac{5}{34} a + \frac{1}{17}$, $\frac{1}{34} a^{10} - \frac{2}{17} a^{7} + \frac{1}{34} a^{6} - \frac{2}{17} a^{5} - \frac{9}{34} a^{4} - \frac{9}{34} a^{3} - \frac{7}{34} a^{2} - \frac{1}{17} a - \frac{1}{17}$, $\frac{1}{34} a^{11} - \frac{7}{34} a^{7} - \frac{7}{17} a^{6} + \frac{13}{34} a^{5} - \frac{1}{34} a^{4} + \frac{5}{34} a^{3} - \frac{6}{17} a^{2} + \frac{3}{17} a + \frac{2}{17}$, $\frac{1}{442} a^{12} - \frac{1}{221} a^{11} - \frac{5}{442} a^{10} + \frac{5}{442} a^{9} + \frac{2}{221} a^{8} + \frac{38}{221} a^{7} + \frac{31}{221} a^{6} - \frac{30}{221} a^{5} - \frac{197}{442} a^{4} - \frac{95}{221} a^{3} + \frac{133}{442} a^{2} - \frac{1}{221} a - \frac{11}{442}$, $\frac{1}{442} a^{13} + \frac{2}{221} a^{11} - \frac{5}{442} a^{10} + \frac{1}{442} a^{9} + \frac{3}{221} a^{8} - \frac{189}{442} a^{7} - \frac{20}{221} a^{6} - \frac{83}{442} a^{5} - \frac{181}{442} a^{4} + \frac{7}{34} a^{3} + \frac{121}{442} a^{2} - \frac{79}{221} a - \frac{37}{221}$, $\frac{1}{442} a^{14} + \frac{3}{442} a^{11} - \frac{5}{442} a^{10} - \frac{1}{442} a^{9} + \frac{3}{442} a^{8} - \frac{94}{221} a^{7} - \frac{97}{442} a^{6} + \frac{75}{221} a^{5} + \frac{73}{442} a^{4} - \frac{47}{221} a^{3} - \frac{183}{442} a^{2} + \frac{25}{442} a - \frac{56}{221}$, $\frac{1}{8579072159405774243651310197854} a^{15} - \frac{2995826423899872404633398184}{4289536079702887121825655098927} a^{14} + \frac{4325971137302164171396164570}{4289536079702887121825655098927} a^{13} + \frac{187311072481703619754990976}{252325651747228654225038535231} a^{12} - \frac{53493166676817241495751996419}{4289536079702887121825655098927} a^{11} - \frac{4183775121400221025837615277}{4289536079702887121825655098927} a^{10} - \frac{29399967338473872561642868643}{8579072159405774243651310197854} a^{9} + \frac{81468610495124963746772380955}{8579072159405774243651310197854} a^{8} + \frac{315534557047225261162196447271}{4289536079702887121825655098927} a^{7} + \frac{415820604483479269440525534782}{4289536079702887121825655098927} a^{6} - \frac{3335916341077146231236193209617}{8579072159405774243651310197854} a^{5} - \frac{2010269834464899687546626428713}{8579072159405774243651310197854} a^{4} + \frac{3984251695355774562594758206777}{8579072159405774243651310197854} a^{3} + \frac{1808377553161642756105712334520}{4289536079702887121825655098927} a^{2} + \frac{2078132645756146978545765130879}{4289536079702887121825655098927} a - \frac{1181212970060261447306875233819}{4289536079702887121825655098927}$

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

Class group and class number

$C_{104}$, which has order $104$ (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:  \( 145944.794365 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-323}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{17}, \sqrt{-19})\), 4.4.4913.1, 4.0.1773593.2, 8.0.3145632129649.3, 8.4.148132260953.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
$19$19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$