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Properties

Label 16.0.51584658396...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{6}\cdot 149^{2}$
Root discriminant $14.78$
Ramified primes $5, 29, 149$
Class number $1$
Class group Trivial
Galois group 16T1445

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 11, -31, 66, -98, 129, -159, 208, -253, 267, -224, 149, -75, 28, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 28*x^14 - 75*x^13 + 149*x^12 - 224*x^11 + 267*x^10 - 253*x^9 + 208*x^8 - 159*x^7 + 129*x^6 - 98*x^5 + 66*x^4 - 31*x^3 + 11*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 7*x^15 + 28*x^14 - 75*x^13 + 149*x^12 - 224*x^11 + 267*x^10 - 253*x^9 + 208*x^8 - 159*x^7 + 129*x^6 - 98*x^5 + 66*x^4 - 31*x^3 + 11*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 28 x^{14} - 75 x^{13} + 149 x^{12} - 224 x^{11} + 267 x^{10} - 253 x^{9} + 208 x^{8} - 159 x^{7} + 129 x^{6} - 98 x^{5} + 66 x^{4} - 31 x^{3} + 11 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5158465839656640625=5^{8}\cdot 29^{6}\cdot 149^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.78$
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:  $5, 29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{22667} a^{15} + \frac{10863}{22667} a^{14} + \frac{8435}{22667} a^{13} + \frac{360}{22667} a^{12} - \frac{8042}{22667} a^{11} + \frac{9855}{22667} a^{10} - \frac{125}{22667} a^{9} + \frac{1017}{22667} a^{8} - \frac{342}{1193} a^{7} - \frac{3047}{22667} a^{6} - \frac{4274}{22667} a^{5} + \frac{8872}{22667} a^{4} - \frac{9379}{22667} a^{3} + \frac{6405}{22667} a^{2} - \frac{10663}{22667} a - \frac{10441}{22667}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 440.9151114 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1445:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1445 are not computed
Character table for t16n1445 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.78318125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$149$149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$