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Properties

Label 16.0.75810861054...6081.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{4}\cdot 61^{12}$
Root discriminant $41.45$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_4$ (as 16T157)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![449, -2897, 7675, -10033, 6786, -4185, 6269, -7364, 6510, -4154, 1834, -516, 86, -28, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 + 86*x^12 - 516*x^11 + 1834*x^10 - 4154*x^9 + 6510*x^8 - 7364*x^7 + 6269*x^6 - 4185*x^5 + 6786*x^4 - 10033*x^3 + 7675*x^2 - 2897*x + 449)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 28*x^13 + 86*x^12 - 516*x^11 + 1834*x^10 - 4154*x^9 + 6510*x^8 - 7364*x^7 + 6269*x^6 - 4185*x^5 + 6786*x^4 - 10033*x^3 + 7675*x^2 - 2897*x + 449, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 28 x^{13} + 86 x^{12} - 516 x^{11} + 1834 x^{10} - 4154 x^{9} + 6510 x^{8} - 7364 x^{7} + 6269 x^{6} - 4185 x^{5} + 6786 x^{4} - 10033 x^{3} + 7675 x^{2} - 2897 x + 449 \)

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:  \(75810861054913358267806081=13^{4}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.45$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} - \frac{6}{13} a^{11} - \frac{6}{13} a^{10} - \frac{6}{13} a^{9} - \frac{3}{13} a^{8} - \frac{5}{13} a^{7} - \frac{1}{13} a^{6} - \frac{5}{13} a^{5} + \frac{1}{13} a^{3} + \frac{3}{13} a^{2} + \frac{1}{13} a + \frac{4}{13}$, $\frac{1}{13} a^{13} - \frac{3}{13} a^{11} - \frac{3}{13} a^{10} + \frac{3}{13} a^{8} - \frac{5}{13} a^{7} + \frac{2}{13} a^{6} - \frac{4}{13} a^{5} + \frac{1}{13} a^{4} - \frac{4}{13} a^{3} + \frac{6}{13} a^{2} - \frac{3}{13} a - \frac{2}{13}$, $\frac{1}{9517933087} a^{14} - \frac{7}{9517933087} a^{13} + \frac{43791591}{9517933087} a^{12} - \frac{262749455}{9517933087} a^{11} - \frac{2205657952}{9517933087} a^{10} + \frac{3918893177}{9517933087} a^{9} + \frac{847011562}{9517933087} a^{8} - \frac{1237849768}{9517933087} a^{7} + \frac{1692229324}{9517933087} a^{6} - \frac{3285864836}{9517933087} a^{5} - \frac{4374428485}{9517933087} a^{4} - \frac{4433378810}{9517933087} a^{3} - \frac{350727923}{732148699} a^{2} + \frac{4339533570}{9517933087} a + \frac{1897835078}{9517933087}$, $\frac{1}{789988446221} a^{15} + \frac{34}{789988446221} a^{14} - \frac{16063480074}{789988446221} a^{13} - \frac{15306714301}{789988446221} a^{12} - \frac{53978712751}{789988446221} a^{11} - \frac{213174807782}{789988446221} a^{10} - \frac{298999899852}{789988446221} a^{9} + \frac{235562665198}{789988446221} a^{8} + \frac{315550440938}{789988446221} a^{7} - \frac{329996908711}{789988446221} a^{6} + \frac{190372027789}{789988446221} a^{5} - \frac{85677021029}{789988446221} a^{4} + \frac{99209998401}{789988446221} a^{3} - \frac{244098940105}{789988446221} a^{2} + \frac{277926637114}{789988446221} a - \frac{338049222834}{789988446221}$

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:  $7$
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:  \( 1356422.33222 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

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

\(\Q(\sqrt{61}) \), 4.0.226981.1, 8.0.669764866693.1 x2, 8.0.8706943267009.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$