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Properties

Label 16.16.6124178019...5808.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 17^{15}\cdot 67^{4}\cdot 63443^{4}$
Root discriminant $1293.27$
Ramified primes $2, 17, 67, 63443$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5549872123325448369911507057, 0, -25569713573708678886123648, 0, 40636889264751328818438, 0, -31542985045966774416, 0, 13470632027977447, 0, -3295638200574, 0, 458180345, 0, -33541, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 33541*x^14 + 458180345*x^12 - 3295638200574*x^10 + 13470632027977447*x^8 - 31542985045966774416*x^6 + 40636889264751328818438*x^4 - 25569713573708678886123648*x^2 + 5549872123325448369911507057)
 
gp: K = bnfinit(x^16 - 33541*x^14 + 458180345*x^12 - 3295638200574*x^10 + 13470632027977447*x^8 - 31542985045966774416*x^6 + 40636889264751328818438*x^4 - 25569713573708678886123648*x^2 + 5549872123325448369911507057, 1)
 

Normalized defining polynomial

\( x^{16} - 33541 x^{14} + 458180345 x^{12} - 3295638200574 x^{10} + 13470632027977447 x^{8} - 31542985045966774416 x^{6} + 40636889264751328818438 x^{4} - 25569713573708678886123648 x^{2} + 5549872123325448369911507057 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(61241780195042689588946649619585484552657695735808=2^{16}\cdot 17^{15}\cdot 67^{4}\cdot 63443^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1293.27$
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:  $2, 17, 67, 63443$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4250681} a^{10} - \frac{33541}{4250681} a^{8} - \frac{893203}{4250681} a^{6} - \frac{207654}{4250681} a^{4} + \frac{350324}{4250681} a^{2}$, $\frac{1}{4250681} a^{11} - \frac{33541}{4250681} a^{9} - \frac{893203}{4250681} a^{7} - \frac{207654}{4250681} a^{5} + \frac{350324}{4250681} a^{3}$, $\frac{1}{18068288963761} a^{12} - \frac{33541}{18068288963761} a^{10} + \frac{458180345}{18068288963761} a^{8} - \frac{3295638200574}{18068288963761} a^{6} - \frac{8311538988259}{18068288963761} a^{4} + \frac{338029}{4250681} a^{2}$, $\frac{1}{18068288963761} a^{13} - \frac{33541}{18068288963761} a^{11} + \frac{458180345}{18068288963761} a^{9} - \frac{3295638200574}{18068288963761} a^{7} - \frac{8311538988259}{18068288963761} a^{5} + \frac{338029}{4250681} a^{3}$, $\frac{1}{1293256621706517252764172204308778130112403369332293027541397386822212777} a^{14} + \frac{32498171672417826243307834961492351962137507088901146940000}{1293256621706517252764172204308778130112403369332293027541397386822212777} a^{12} + \frac{90077709058272898886901712319434449252588646519723314957681937481}{1293256621706517252764172204308778130112403369332293027541397386822212777} a^{10} - \frac{248352335175671100782108845609246256565569751716604053303923159186463875}{1293256621706517252764172204308778130112403369332293027541397386822212777} a^{8} - \frac{308068725747858734350239290577984719809437181088723428440535630190925436}{1293256621706517252764172204308778130112403369332293027541397386822212777} a^{6} + \frac{53349460294104322383922547515734469221739344710964811486757942243}{304246924600203415114936219469016407044519071022335721627051615217} a^{4} - \frac{13702212150867952903315572616759974245540461494700287620991}{71576042662388312629184881074118807561545801960282533934457} a^{2} - \frac{4883424118840731326850425282450502888222424584306429}{16838723645079062067745116858714828885429370484466497}$, $\frac{1}{1293256621706517252764172204308778130112403369332293027541397386822212777} a^{15} + \frac{32498171672417826243307834961492351962137507088901146940000}{1293256621706517252764172204308778130112403369332293027541397386822212777} a^{13} + \frac{90077709058272898886901712319434449252588646519723314957681937481}{1293256621706517252764172204308778130112403369332293027541397386822212777} a^{11} - \frac{248352335175671100782108845609246256565569751716604053303923159186463875}{1293256621706517252764172204308778130112403369332293027541397386822212777} a^{9} - \frac{308068725747858734350239290577984719809437181088723428440535630190925436}{1293256621706517252764172204308778130112403369332293027541397386822212777} a^{7} + \frac{53349460294104322383922547515734469221739344710964811486757942243}{304246924600203415114936219469016407044519071022335721627051615217} a^{5} - \frac{13702212150867952903315572616759974245540461494700287620991}{71576042662388312629184881074118807561545801960282533934457} a^{3} - \frac{4883424118840731326850425282450502888222424584306429}{16838723645079062067745116858714828885429370484466497} a$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $15$
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:  \( 11999966274600000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 R $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
17Data not computed
67Data not computed
63443Data not computed