Properties

Label 16.0.4559007230078125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 11^{4}\cdot 59^{2}\cdot 229$
Root discriminant $9.52$
Ramified primes $5, 11, 59, 229$
Class number $1$
Class group Trivial
Galois Group 16T1823

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 3, -6, 11, -19, 26, -30, 33, -30, 26, -19, 11, -6, 3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 6*x^13 + 11*x^12 - 19*x^11 + 26*x^10 - 30*x^9 + 33*x^8 - 30*x^7 + 26*x^6 - 19*x^5 + 11*x^4 - 6*x^3 + 3*x^2 - 2*x + 1)
gp: K = bnfinit(x^16 - 2*x^15 + 3*x^14 - 6*x^13 + 11*x^12 - 19*x^11 + 26*x^10 - 30*x^9 + 33*x^8 - 30*x^7 + 26*x^6 - 19*x^5 + 11*x^4 - 6*x^3 + 3*x^2 - 2*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 2 x^{15} \) \(\mathstrut +\mathstrut 3 x^{14} \) \(\mathstrut -\mathstrut 6 x^{13} \) \(\mathstrut +\mathstrut 11 x^{12} \) \(\mathstrut -\mathstrut 19 x^{11} \) \(\mathstrut +\mathstrut 26 x^{10} \) \(\mathstrut -\mathstrut 30 x^{9} \) \(\mathstrut +\mathstrut 33 x^{8} \) \(\mathstrut -\mathstrut 30 x^{7} \) \(\mathstrut +\mathstrut 26 x^{6} \) \(\mathstrut -\mathstrut 19 x^{5} \) \(\mathstrut +\mathstrut 11 x^{4} \) \(\mathstrut -\mathstrut 6 x^{3} \) \(\mathstrut +\mathstrut 3 x^{2} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(4559007230078125=5^{8}\cdot 11^{4}\cdot 59^{2}\cdot 229\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.52$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 11, 59, 229$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{199} a^{14} + \frac{34}{199} a^{13} + \frac{32}{199} a^{12} - \frac{82}{199} a^{11} + \frac{12}{199} a^{10} + \frac{97}{199} a^{9} - \frac{76}{199} a^{8} - \frac{77}{199} a^{7} - \frac{76}{199} a^{6} + \frac{97}{199} a^{5} + \frac{12}{199} a^{4} - \frac{82}{199} a^{3} + \frac{32}{199} a^{2} + \frac{34}{199} a + \frac{1}{199}$, $\frac{1}{199} a^{15} + \frac{70}{199} a^{13} + \frac{24}{199} a^{12} + \frac{14}{199} a^{11} + \frac{87}{199} a^{10} + \frac{9}{199} a^{9} - \frac{80}{199} a^{8} - \frac{45}{199} a^{7} + \frac{94}{199} a^{6} + \frac{97}{199} a^{5} - \frac{92}{199} a^{4} + \frac{34}{199} a^{3} - \frac{59}{199} a^{2} + \frac{39}{199} a - \frac{34}{199}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( a \),  \( \frac{107}{199} a^{15} + \frac{10}{199} a^{14} + \frac{69}{199} a^{13} - \frac{296}{199} a^{12} + \frac{280}{199} a^{11} - \frac{521}{199} a^{10} + \frac{341}{199} a^{9} - \frac{365}{199} a^{8} + \frac{584}{199} a^{7} - \frac{55}{199} a^{6} + \frac{603}{199} a^{5} - \frac{172}{199} a^{4} + \frac{231}{199} a^{3} - \frac{222}{199} a^{2} + \frac{135}{199} a - \frac{46}{199} \),  \( \frac{195}{199} a^{15} - \frac{334}{199} a^{14} + \frac{503}{199} a^{13} - \frac{1033}{199} a^{12} + \frac{1860}{199} a^{11} - \frac{3162}{199} a^{10} + \frac{4182}{199} a^{9} - \frac{4743}{199} a^{8} + \frac{5003}{199} a^{7} - \frac{4245}{199} a^{6} + \frac{3631}{199} a^{5} - \frac{2446}{199} a^{4} + \frac{1382}{199} a^{3} - \frac{502}{199} a^{2} + \frac{229}{199} a - \frac{198}{199} \),  \( \frac{12}{199} a^{15} - \frac{103}{199} a^{14} + \frac{124}{199} a^{13} - \frac{222}{199} a^{12} + \frac{455}{199} a^{11} - \frac{789}{199} a^{10} + \frac{1261}{199} a^{9} - \frac{1490}{199} a^{8} + \frac{1620}{199} a^{7} - \frac{1591}{199} a^{6} + \frac{1322}{199} a^{5} - \frac{1146}{199} a^{4} + \frac{496}{199} a^{3} - \frac{223}{199} a^{2} - \frac{49}{199} a - \frac{113}{199} \),  \( \frac{82}{199} a^{15} - \frac{118}{199} a^{14} + \frac{136}{199} a^{13} - \frac{415}{199} a^{12} + \frac{675}{199} a^{11} - \frac{1048}{199} a^{10} + \frac{1431}{199} a^{9} - \frac{1572}{199} a^{8} + \frac{1814}{199} a^{7} - \frac{1433}{199} a^{6} + \frac{1284}{199} a^{5} - \frac{1199}{199} a^{4} + \frac{524}{199} a^{3} - \frac{455}{199} a^{2} + \frac{181}{199} a - \frac{120}{199} \),  \( \frac{4}{199} a^{15} - \frac{88}{199} a^{14} + \frac{74}{199} a^{13} - \frac{133}{199} a^{12} + \frac{307}{199} a^{11} - \frac{509}{199} a^{10} + \frac{853}{199} a^{9} - 4 a^{8} + \frac{825}{199} a^{7} - \frac{896}{199} a^{6} + \frac{409}{199} a^{5} - \frac{429}{199} a^{4} - \frac{210}{199} a^{3} + \frac{132}{199} a^{2} - \frac{50}{199} a + \frac{174}{199} \),  \( \frac{87}{199} a^{15} - \frac{123}{199} a^{14} + \frac{117}{199} a^{13} - \frac{256}{199} a^{12} + \frac{558}{199} a^{11} - \frac{872}{199} a^{10} + \frac{792}{199} a^{9} - 3 a^{8} + \frac{382}{199} a^{7} + \frac{14}{199} a^{6} - \frac{109}{199} a^{5} + \frac{470}{199} a^{4} - \frac{289}{199} a^{3} + \frac{284}{199} a^{2} + \frac{7}{199} a - \frac{96}{199} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 9.23010708309 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

16T1823:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 32768
The 230 conjugacy class representatives for t16n1823 are not computed
Character table for t16n1823 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.2.4461875.1

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
229Data not computed