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Properties

Label 16.16.5002464129...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{12}\cdot 5^{12}\cdot 29^{8}$
Root discriminant $30.28$
Ramified primes $2, 5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_8:C_2$ (as 16T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 11, -23, -402, -1033, -259, 2227, 2474, -620, -2139, -641, 483, 257, -28, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 28*x^14 - 28*x^13 + 257*x^12 + 483*x^11 - 641*x^10 - 2139*x^9 - 620*x^8 + 2474*x^7 + 2227*x^6 - 259*x^5 - 1033*x^4 - 402*x^3 - 23*x^2 + 11*x + 1)
 
gp: K = bnfinit(x^16 - 28*x^14 - 28*x^13 + 257*x^12 + 483*x^11 - 641*x^10 - 2139*x^9 - 620*x^8 + 2474*x^7 + 2227*x^6 - 259*x^5 - 1033*x^4 - 402*x^3 - 23*x^2 + 11*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 28 x^{14} - 28 x^{13} + 257 x^{12} + 483 x^{11} - 641 x^{10} - 2139 x^{9} - 620 x^{8} + 2474 x^{7} + 2227 x^{6} - 259 x^{5} - 1033 x^{4} - 402 x^{3} - 23 x^{2} + 11 x + 1 \)

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:  \(500246412961000000000000=2^{12}\cdot 5^{12}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.28$
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:  $2, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{431} a^{14} + \frac{75}{431} a^{13} - \frac{137}{431} a^{12} + \frac{129}{431} a^{11} - \frac{136}{431} a^{10} + \frac{106}{431} a^{9} + \frac{127}{431} a^{8} - \frac{35}{431} a^{7} - \frac{56}{431} a^{6} - \frac{158}{431} a^{5} - \frac{132}{431} a^{4} + \frac{195}{431} a^{3} - \frac{148}{431} a^{2} + \frac{19}{431} a + \frac{102}{431}$, $\frac{1}{939149} a^{15} + \frac{727}{939149} a^{14} + \frac{94880}{939149} a^{13} - \frac{170654}{939149} a^{12} + \frac{142157}{939149} a^{11} + \frac{454063}{939149} a^{10} + \frac{267930}{939149} a^{9} + \frac{433603}{939149} a^{8} + \frac{14621}{939149} a^{7} + \frac{451653}{939149} a^{6} + \frac{455859}{939149} a^{5} - \frac{156122}{939149} a^{4} + \frac{85185}{939149} a^{3} - \frac{87426}{939149} a^{2} - \frac{321966}{939149} a - \frac{7628}{939149}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 3348865.4945 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8:C_2$ (as 16T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.5$x^{8} + 6 x^{6} + 8 x^{5} + 80$$2$$4$$12$$C_8$$[3]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$