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Properties

Label 16.0.16821502541...4417.2
Degree $16$
Signature $[0, 8]$
Discriminant $17^{13}\cdot 19^{8}$
Root discriminant $43.56$
Ramified primes $17, 19$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4489, -17956, 48763, -57267, 55145, -41377, 29961, -15630, 9760, -4138, 1316, -118, -122, 74, -10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 10*x^14 + 74*x^13 - 122*x^12 - 118*x^11 + 1316*x^10 - 4138*x^9 + 9760*x^8 - 15630*x^7 + 29961*x^6 - 41377*x^5 + 55145*x^4 - 57267*x^3 + 48763*x^2 - 17956*x + 4489)
 
gp: K = bnfinit(x^16 - 4*x^15 - 10*x^14 + 74*x^13 - 122*x^12 - 118*x^11 + 1316*x^10 - 4138*x^9 + 9760*x^8 - 15630*x^7 + 29961*x^6 - 41377*x^5 + 55145*x^4 - 57267*x^3 + 48763*x^2 - 17956*x + 4489, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 10 x^{14} + 74 x^{13} - 122 x^{12} - 118 x^{11} + 1316 x^{10} - 4138 x^{9} + 9760 x^{8} - 15630 x^{7} + 29961 x^{6} - 41377 x^{5} + 55145 x^{4} - 57267 x^{3} + 48763 x^{2} - 17956 x + 4489 \)

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:  \(168215025416361753462674417=17^{13}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.56$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} + \frac{2}{7} a^{10} + \frac{2}{7} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{11} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7}$, $\frac{1}{49} a^{14} + \frac{1}{49} a^{13} + \frac{1}{49} a^{12} - \frac{13}{49} a^{11} + \frac{15}{49} a^{10} - \frac{23}{49} a^{9} + \frac{17}{49} a^{8} + \frac{23}{49} a^{7} - \frac{19}{49} a^{6} - \frac{5}{49} a^{5} - \frac{19}{49} a^{4} + \frac{1}{49} a^{3} + \frac{9}{49} a^{2} + \frac{16}{49} a - \frac{5}{49}$, $\frac{1}{3154745737613069898853529219066227} a^{15} - \frac{15276625598969884329641237900226}{3154745737613069898853529219066227} a^{14} + \frac{73278632594213190729092755833182}{3154745737613069898853529219066227} a^{13} + \frac{164978235466120965356439929224850}{3154745737613069898853529219066227} a^{12} - \frac{406817830570640617689334102182761}{3154745737613069898853529219066227} a^{11} + \frac{444276706259400710419614799464548}{3154745737613069898853529219066227} a^{10} + \frac{569902809305207807485410289328212}{3154745737613069898853529219066227} a^{9} + \frac{563169004742187025196503436177214}{3154745737613069898853529219066227} a^{8} - \frac{178222745667905393209509811163222}{3154745737613069898853529219066227} a^{7} - \frac{322991572935996359380736044747211}{3154745737613069898853529219066227} a^{6} - \frac{500474023812124586530042883291862}{3154745737613069898853529219066227} a^{5} + \frac{609389961077300985513447208431679}{3154745737613069898853529219066227} a^{4} + \frac{567197440872882653016663200510303}{3154745737613069898853529219066227} a^{3} + \frac{1074692115448495593552556894427101}{3154745737613069898853529219066227} a^{2} + \frac{268891314845063627357644277421690}{3154745737613069898853529219066227} a - \frac{6958752624070584335634026067400}{47085757277807013415724316702481}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 1119942.74184 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.6137.1, 8.0.185037184097.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$17.8.7.5$x^{8} + 459$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.6.3$x^{8} - 17 x^{4} + 867$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
19Data not computed