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Properties

Label 16.0.29600926661...0129.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{10}\cdot 59^{8}$
Root discriminant $45.13$
Ramified primes $17, 59$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1787, -820, 3509, -6406, 6157, -4546, 2946, -696, -623, 766, -426, 180, -30, -28, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 - 30*x^12 + 180*x^11 - 426*x^10 + 766*x^9 - 623*x^8 - 696*x^7 + 2946*x^6 - 4546*x^5 + 6157*x^4 - 6406*x^3 + 3509*x^2 - 820*x + 1787)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 28*x^13 - 30*x^12 + 180*x^11 - 426*x^10 + 766*x^9 - 623*x^8 - 696*x^7 + 2946*x^6 - 4546*x^5 + 6157*x^4 - 6406*x^3 + 3509*x^2 - 820*x + 1787, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 28 x^{13} - 30 x^{12} + 180 x^{11} - 426 x^{10} + 766 x^{9} - 623 x^{8} - 696 x^{7} + 2946 x^{6} - 4546 x^{5} + 6157 x^{4} - 6406 x^{3} + 3509 x^{2} - 820 x + 1787 \)

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:  \(296009266610568616126240129=17^{10}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.13$
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:  $17, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{3} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{270} a^{13} + \frac{8}{135} a^{12} + \frac{7}{135} a^{11} + \frac{17}{270} a^{10} - \frac{11}{54} a^{9} - \frac{11}{45} a^{8} + \frac{37}{90} a^{7} - \frac{16}{45} a^{6} + \frac{91}{270} a^{5} - \frac{7}{18} a^{4} + \frac{89}{270} a^{3} - \frac{1}{10} a^{2} + \frac{19}{135} a - \frac{52}{135}$, $\frac{1}{2019875130} a^{14} - \frac{1}{288553590} a^{13} - \frac{14256103}{673291710} a^{12} - \frac{8003591}{201987513} a^{11} + \frac{26384653}{155375010} a^{10} - \frac{27510181}{2019875130} a^{9} - \frac{7189813}{74810190} a^{8} - \frac{102020924}{336645855} a^{7} + \frac{796630099}{2019875130} a^{6} - \frac{397758889}{1009937565} a^{5} + \frac{33104872}{1009937565} a^{4} - \frac{65155277}{1009937565} a^{3} - \frac{134617843}{288553590} a^{2} - \frac{21450491}{673291710} a - \frac{15606173}{2019875130}$, $\frac{1}{159570135270} a^{15} + \frac{16}{79785067635} a^{14} + \frac{53072977}{53190045090} a^{13} + \frac{5286619166}{79785067635} a^{12} + \frac{262760287}{12274625790} a^{11} - \frac{8945613227}{79785067635} a^{10} - \frac{827741062}{5319004509} a^{9} + \frac{5335419409}{26595022545} a^{8} + \frac{30643592573}{79785067635} a^{7} - \frac{7249632973}{31914027054} a^{6} - \frac{11672391239}{31914027054} a^{5} - \frac{7296403069}{22795733610} a^{4} + \frac{2811145319}{15957013527} a^{3} + \frac{650190687}{1970001670} a^{2} + \frac{69449570179}{159570135270} a + \frac{1482899479}{4091541930}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 1476026.95846 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{-59}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1003}) \), 4.0.59177.1 x2, 4.2.17051.1 x2, \(\Q(\sqrt{17}, \sqrt{-59})\), 8.0.59532594593.1, 8.0.17204919837377.1, 8.0.1012054108081.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ 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
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$59$59.8.4.1$x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
59.8.4.1$x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$