Properties

Label 16.8.252...625.1
Degree $16$
Signature $[8, 4]$
Discriminant $2.528\times 10^{18}$
Root discriminant $14.13$
Ramified primes $5, 11, 29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_4^2.C_2$ (as 16T388)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 6*x^14 + 17*x^13 - 68*x^12 + 84*x^11 + 22*x^10 - 225*x^9 + 352*x^8 - 237*x^7 - 82*x^6 + 347*x^5 - 379*x^4 + 251*x^3 - 92*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 6*x^14 + 17*x^13 - 68*x^12 + 84*x^11 + 22*x^10 - 225*x^9 + 352*x^8 - 237*x^7 - 82*x^6 + 347*x^5 - 379*x^4 + 251*x^3 - 92*x^2 + 8*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, -92, 251, -379, 347, -82, -237, 352, -225, 22, 84, -68, 17, 6, -5, 1]);
 

\(x^{16} - 5 x^{15} + 6 x^{14} + 17 x^{13} - 68 x^{12} + 84 x^{11} + 22 x^{10} - 225 x^{9} + 352 x^{8} - 237 x^{7} - 82 x^{6} + 347 x^{5} - 379 x^{4} + 251 x^{3} - 92 x^{2} + 8 x + 1\)  Toggle raw display

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2528149687744140625\)\(\medspace = 5^{12}\cdot 11^{4}\cdot 29^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $14.13$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 11, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5363777675} a^{15} + \frac{322937803}{5363777675} a^{14} + \frac{12500714}{1072755535} a^{13} + \frac{492210692}{5363777675} a^{12} - \frac{412600122}{5363777675} a^{11} - \frac{995742422}{5363777675} a^{10} - \frac{1961080849}{5363777675} a^{9} + \frac{1010963983}{5363777675} a^{8} - \frac{26072708}{233207725} a^{7} + \frac{362739456}{5363777675} a^{6} + \frac{1115524371}{5363777675} a^{5} + \frac{483742931}{1072755535} a^{4} + \frac{582791341}{5363777675} a^{3} + \frac{1802964564}{5363777675} a^{2} - \frac{304876896}{1072755535} a - \frac{2597824497}{5363777675}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1156.04018319 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 1156.04018319 \cdot 1}{2\sqrt{2528149687744140625}}\approx 0.145044405090$ (assuming GRH)

Galois group

$D_4^2.C_2$ (as 16T388):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 128
The 20 conjugacy class representatives for $D_4^2.C_2$
Character table for $D_4^2.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 4.4.725.1, 4.2.7975.1, 8.4.54828125.1 x2, 8.6.144546875.1 x2, 8.4.63600625.2

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 ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ R ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ R ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$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}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$