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Properties

Label 16.8.19349710423...1936.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 17^{14}\cdot 67^{2}$
Root discriminant $28.54$
Ramified primes $2, 17, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![239, 705, 505, -124, -817, -1090, -201, 233, 317, -75, -22, 118, -101, 24, 3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 3*x^14 + 24*x^13 - 101*x^12 + 118*x^11 - 22*x^10 - 75*x^9 + 317*x^8 + 233*x^7 - 201*x^6 - 1090*x^5 - 817*x^4 - 124*x^3 + 505*x^2 + 705*x + 239)
 
gp: K = bnfinit(x^16 - 4*x^15 + 3*x^14 + 24*x^13 - 101*x^12 + 118*x^11 - 22*x^10 - 75*x^9 + 317*x^8 + 233*x^7 - 201*x^6 - 1090*x^5 - 817*x^4 - 124*x^3 + 505*x^2 + 705*x + 239, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 3 x^{14} + 24 x^{13} - 101 x^{12} + 118 x^{11} - 22 x^{10} - 75 x^{9} + 317 x^{8} + 233 x^{7} - 201 x^{6} - 1090 x^{5} - 817 x^{4} - 124 x^{3} + 505 x^{2} + 705 x + 239 \)

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:  \(193497104236838597191936=2^{8}\cdot 17^{14}\cdot 67^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.54$
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:  $2, 17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{34} a^{12} - \frac{1}{2} a^{11} + \frac{7}{17} a^{10} + \frac{8}{17} a^{9} + \frac{7}{17} a^{8} + \frac{15}{34} a^{7} - \frac{13}{34} a^{6} + \frac{5}{17} a^{5} + \frac{5}{34} a^{4} + \frac{7}{34} a^{3} + \frac{3}{34} a^{2} - \frac{5}{34} a + \frac{1}{34}$, $\frac{1}{34} a^{13} - \frac{3}{34} a^{11} + \frac{8}{17} a^{10} + \frac{7}{17} a^{9} + \frac{15}{34} a^{8} + \frac{2}{17} a^{7} - \frac{7}{34} a^{6} + \frac{5}{34} a^{5} - \frac{5}{17} a^{4} - \frac{7}{17} a^{3} + \frac{6}{17} a^{2} - \frac{8}{17} a - \frac{1}{2}$, $\frac{1}{34} a^{14} - \frac{1}{34} a^{11} - \frac{6}{17} a^{10} - \frac{5}{34} a^{9} + \frac{6}{17} a^{8} + \frac{2}{17} a^{7} - \frac{7}{17} a^{5} + \frac{1}{34} a^{4} - \frac{1}{34} a^{3} - \frac{7}{34} a^{2} + \frac{1}{17} a + \frac{3}{34}$, $\frac{1}{12555781018540274722} a^{15} + \frac{39380938527320423}{6277890509270137361} a^{14} - \frac{111794022958083609}{12555781018540274722} a^{13} - \frac{6177830576578694}{6277890509270137361} a^{12} - \frac{879975783298591319}{6277890509270137361} a^{11} - \frac{4573004859447069135}{12555781018540274722} a^{10} + \frac{17796176557918059}{6277890509270137361} a^{9} - \frac{334342432629244011}{738575354031780866} a^{8} - \frac{5054084117961286885}{12555781018540274722} a^{7} - \frac{1510844948951584789}{6277890509270137361} a^{6} - \frac{3018677973449342975}{6277890509270137361} a^{5} - \frac{149969881244088993}{369287677015890433} a^{4} + \frac{66567637565851991}{369287677015890433} a^{3} + \frac{4897584007943236745}{12555781018540274722} a^{2} + \frac{1106817164590274928}{6277890509270137361} a + \frac{2146343647338065981}{6277890509270137361}$

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:  \( 462901.867367 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
67Data not computed