↑

Properties

Label 16.0.40073498016...161.17
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 53^{12}$
Root discriminant $70.82$
Ramified primes $13, 53$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41111, -97055, 221734, -192733, 189151, -42762, 79, 9223, -10939, 1863, 2336, -480, -39, -37, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 9*x^14 - 37*x^13 - 39*x^12 - 480*x^11 + 2336*x^10 + 1863*x^9 - 10939*x^8 + 9223*x^7 + 79*x^6 - 42762*x^5 + 189151*x^4 - 192733*x^3 + 221734*x^2 - 97055*x + 41111)
 
gp: K = bnfinit(x^16 - 3*x^15 + 9*x^14 - 37*x^13 - 39*x^12 - 480*x^11 + 2336*x^10 + 1863*x^9 - 10939*x^8 + 9223*x^7 + 79*x^6 - 42762*x^5 + 189151*x^4 - 192733*x^3 + 221734*x^2 - 97055*x + 41111, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 9 x^{14} - 37 x^{13} - 39 x^{12} - 480 x^{11} + 2336 x^{10} + 1863 x^{9} - 10939 x^{8} + 9223 x^{7} + 79 x^{6} - 42762 x^{5} + 189151 x^{4} - 192733 x^{3} + 221734 x^{2} - 97055 x + 41111 \)

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:  \(400734980167009195224860426161=13^{8}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.82$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{3}{13} a^{11} + \frac{2}{13} a^{10} + \frac{4}{13} a^{8} - \frac{6}{13} a^{7} + \frac{5}{13} a^{6} - \frac{1}{13} a^{5} - \frac{5}{13} a^{4} + \frac{1}{13} a^{3} - \frac{2}{13} a^{2} + \frac{3}{13} a - \frac{5}{13}$, $\frac{1}{13} a^{13} + \frac{6}{13} a^{11} - \frac{6}{13} a^{10} + \frac{4}{13} a^{9} - \frac{5}{13} a^{8} - \frac{3}{13} a^{7} - \frac{3}{13} a^{6} - \frac{2}{13} a^{5} + \frac{3}{13} a^{4} - \frac{5}{13} a^{3} - \frac{4}{13} a^{2} - \frac{1}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{14} + \frac{2}{13} a^{11} + \frac{5}{13} a^{10} - \frac{5}{13} a^{9} - \frac{1}{13} a^{8} - \frac{6}{13} a^{7} - \frac{6}{13} a^{6} - \frac{4}{13} a^{5} - \frac{1}{13} a^{4} + \frac{3}{13} a^{3} - \frac{2}{13} a^{2} - \frac{3}{13} a + \frac{4}{13}$, $\frac{1}{677514639370206736916259267191925777127} a^{15} - \frac{20088325370520168265846423691137914795}{677514639370206736916259267191925777127} a^{14} - \frac{8701689389912148338688534003503345940}{677514639370206736916259267191925777127} a^{13} - \frac{127183050135990955462262461803621332}{4547078116578568704135968236187421323} a^{12} + \frac{273856240282944412786359771316099806649}{677514639370206736916259267191925777127} a^{11} + \frac{223275313986154707122027615576934398366}{677514639370206736916259267191925777127} a^{10} - \frac{40273923103912852306728589961357793914}{677514639370206736916259267191925777127} a^{9} - \frac{248373684685129941017051635495493150558}{677514639370206736916259267191925777127} a^{8} - \frac{166611630000999186898898092998144791588}{677514639370206736916259267191925777127} a^{7} + \frac{251282964991064628780217962378350504956}{677514639370206736916259267191925777127} a^{6} - \frac{50521147004970502643781798042319961575}{677514639370206736916259267191925777127} a^{5} + \frac{156067918357775361112534491628016573599}{677514639370206736916259267191925777127} a^{4} + \frac{179984382211172494151706854096138599224}{677514639370206736916259267191925777127} a^{3} - \frac{255768508289341275956888869780757469604}{677514639370206736916259267191925777127} a^{2} + \frac{157478852747561740621153618899346933118}{677514639370206736916259267191925777127} a + \frac{6040192074788154023119683893566657731}{96787805624315248130894181027417968161}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( 9534876.17637 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.4.36517.1, 4.0.148877.1, 4.0.1935401.1, 8.4.11944081475573.5, 8.4.11944081475573.4, 8.0.3745777030801.2

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$