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Properties

Label 16.0.14816104373...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 41^{14}$
Root discriminant $57.63$
Ramified primes $5, 41$
Class number $54$ (GRH)
Class group $[3, 3, 6]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18419, -70325, 154504, -223845, 244446, -210738, 151269, -89962, 43548, -16579, 4342, -134, -441, 160, -8, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419)
 
gp: K = bnfinit(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 8 x^{14} + 160 x^{13} - 441 x^{12} - 134 x^{11} + 4342 x^{10} - 16579 x^{9} + 43548 x^{8} - 89962 x^{7} + 151269 x^{6} - 210738 x^{5} + 244446 x^{4} - 223845 x^{3} + 154504 x^{2} - 70325 x + 18419 \)

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:  \(14816104373013890157094140625=5^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.63$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{118} a^{14} - \frac{1}{118} a^{13} + \frac{15}{118} a^{12} - \frac{29}{118} a^{11} + \frac{11}{118} a^{10} + \frac{53}{118} a^{9} - \frac{41}{118} a^{8} - \frac{19}{118} a^{7} - \frac{57}{118} a^{6} - \frac{37}{118} a^{5} + \frac{41}{118} a^{4} + \frac{5}{118} a^{3} - \frac{57}{118} a^{2} + \frac{33}{118} a + \frac{27}{118}$, $\frac{1}{31483304674590496943003883029366} a^{15} + \frac{31931026479879894439667697010}{15741652337295248471501941514683} a^{14} + \frac{2157088679110066712125078096882}{15741652337295248471501941514683} a^{13} + \frac{3184112874432841018839849979641}{15741652337295248471501941514683} a^{12} + \frac{4014361578900052439451301113507}{15741652337295248471501941514683} a^{11} - \frac{6854792361587431176397190068751}{15741652337295248471501941514683} a^{10} - \frac{4129264753035362642321199591721}{15741652337295248471501941514683} a^{9} + \frac{703379679805432398563934017377}{15741652337295248471501941514683} a^{8} - \frac{3281431170124349551830660521436}{15741652337295248471501941514683} a^{7} - \frac{1508809416509151386937829772258}{15741652337295248471501941514683} a^{6} - \frac{2291192466140734266712135117753}{15741652337295248471501941514683} a^{5} - \frac{7213099851565010295879036533467}{15741652337295248471501941514683} a^{4} + \frac{4042141206180289241168920952208}{15741652337295248471501941514683} a^{3} + \frac{3735454260999111702451150015873}{15741652337295248471501941514683} a^{2} + \frac{3919056217585378373499113243469}{15741652337295248471501941514683} a + \frac{8389940394755155296363790383649}{31483304674590496943003883029366}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{6}$, which has order $54$ (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:  \( 645179.367444 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.8405.1, 4.4.68921.1, 4.0.344605.1, 8.8.4868856847025.1, 8.0.121721421175625.1, 8.0.118752606025.1

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{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}$
41Data not computed