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Properties

Label 16.0.23705766996...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{4}\cdot 41^{14}$
Root discriminant $38.54$
Ramified primes $5, 41$
Class number $3$ (GRH)
Class group $[3]$ (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![21904, -46472, 54312, -53128, 52463, -44891, 32032, -20192, 11360, -5723, 2589, -1044, 347, -99, 27, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 27*x^14 - 99*x^13 + 347*x^12 - 1044*x^11 + 2589*x^10 - 5723*x^9 + 11360*x^8 - 20192*x^7 + 32032*x^6 - 44891*x^5 + 52463*x^4 - 53128*x^3 + 54312*x^2 - 46472*x + 21904)
 
gp: K = bnfinit(x^16 - 5*x^15 + 27*x^14 - 99*x^13 + 347*x^12 - 1044*x^11 + 2589*x^10 - 5723*x^9 + 11360*x^8 - 20192*x^7 + 32032*x^6 - 44891*x^5 + 52463*x^4 - 53128*x^3 + 54312*x^2 - 46472*x + 21904, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 27 x^{14} - 99 x^{13} + 347 x^{12} - 1044 x^{11} + 2589 x^{10} - 5723 x^{9} + 11360 x^{8} - 20192 x^{7} + 32032 x^{6} - 44891 x^{5} + 52463 x^{4} - 53128 x^{3} + 54312 x^{2} - 46472 x + 21904 \)

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:  \(23705766996822224251350625=5^{4}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.54$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{236} a^{13} - \frac{23}{236} a^{12} + \frac{7}{59} a^{11} + \frac{3}{118} a^{10} - \frac{21}{236} a^{9} - \frac{75}{236} a^{8} + \frac{41}{236} a^{7} + \frac{43}{236} a^{6} - \frac{43}{236} a^{5} - \frac{27}{118} a^{4} + \frac{73}{236} a^{3} + \frac{1}{236} a^{2} - \frac{22}{59} a + \frac{13}{59}$, $\frac{1}{236} a^{14} - \frac{29}{236} a^{12} + \frac{1}{236} a^{11} - \frac{1}{236} a^{10} + \frac{8}{59} a^{9} + \frac{27}{236} a^{8} - \frac{17}{236} a^{7} - \frac{57}{236} a^{6} + \frac{19}{236} a^{5} + \frac{11}{236} a^{4} - \frac{31}{236} a^{3} + \frac{28}{59} a^{2} + \frac{17}{118} a + \frac{4}{59}$, $\frac{1}{12552512468697992302648072} a^{15} - \frac{26124034610302239881453}{12552512468697992302648072} a^{14} + \frac{23846774496839795439181}{12552512468697992302648072} a^{13} + \frac{588683959971492096687679}{12552512468697992302648072} a^{12} - \frac{1260649245992608347274421}{12552512468697992302648072} a^{11} + \frac{64847578240012285784889}{1569064058587249037831009} a^{10} + \frac{622782002725050519370407}{12552512468697992302648072} a^{9} - \frac{607098880777789572202725}{12552512468697992302648072} a^{8} - \frac{1106342197833429483116841}{6276256234348996151324036} a^{7} - \frac{921528737797235735765605}{6276256234348996151324036} a^{6} - \frac{2219452450061124689431981}{6276256234348996151324036} a^{5} - \frac{535261598480967804680495}{12552512468697992302648072} a^{4} + \frac{3382531466489498384429737}{12552512468697992302648072} a^{3} - \frac{2887322447355385113829099}{6276256234348996151324036} a^{2} + \frac{1250648956612332013296221}{3138128117174498075662018} a + \frac{870858816923374359642}{42407136718574298319757}$

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:  \( 322589.683722 \) (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, 8.0.194754273881.1, 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ 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$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
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