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Properties

Label 16.0.3345005787800625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.345\times 10^{15}$
Root discriminant $9.34$
Ramified primes $3, 5, 13$
Class number $1$
Class group trivial
Galois group $D_8:C_2$ (as 16T47)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 12*x^14 - 19*x^13 + 21*x^12 - 16*x^11 + 6*x^10 + 12*x^9 - 28*x^8 + 21*x^7 + x^6 - 20*x^5 + 28*x^4 - 20*x^3 + 10*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 12*x^14 - 19*x^13 + 21*x^12 - 16*x^11 + 6*x^10 + 12*x^9 - 28*x^8 + 21*x^7 + x^6 - 20*x^5 + 28*x^4 - 20*x^3 + 10*x^2 - 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 10, -20, 28, -20, 1, 21, -28, 12, 6, -16, 21, -19, 12, -5, 1]);
 

\( x^{16} - 5 x^{15} + 12 x^{14} - 19 x^{13} + 21 x^{12} - 16 x^{11} + 6 x^{10} + 12 x^{9} - 28 x^{8} + 21 x^{7} + x^{6} - 20 x^{5} + 28 x^{4} - 20 x^{3} + 10 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3345005787800625\)\(\medspace = 3^{8}\cdot 5^{4}\cdot 13^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.34$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4201259} a^{15} - \frac{844160}{4201259} a^{14} + \frac{1138268}{4201259} a^{13} - \frac{476410}{4201259} a^{12} - \frac{1634204}{4201259} a^{11} + \frac{273623}{4201259} a^{10} + \frac{795002}{4201259} a^{9} - \frac{1897}{4201259} a^{8} + \frac{682328}{4201259} a^{7} + \frac{2016081}{4201259} a^{6} - \frac{1049503}{4201259} a^{5} - \frac{1487939}{4201259} a^{4} + \frac{743343}{4201259} a^{3} - \frac{867204}{4201259} a^{2} + \frac{2016916}{4201259} a - \frac{107421}{4201259}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{8947}{8033} a^{15} - \frac{40788}{8033} a^{14} + \frac{87386}{8033} a^{13} - \frac{122437}{8033} a^{12} + \frac{114059}{8033} a^{11} - \frac{64231}{8033} a^{10} - \frac{1220}{8033} a^{9} + \frac{121765}{8033} a^{8} - \frac{194988}{8033} a^{7} + \frac{72428}{8033} a^{6} + \frac{87250}{8033} a^{5} - \frac{166072}{8033} a^{4} + \frac{161088}{8033} a^{3} - \frac{64577}{8033} a^{2} + \frac{24285}{8033} a - \frac{3468}{8033} \) (order $6$)
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 22.9306095521 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 22.9306095521 \cdot 1}{6\sqrt{3345005787800625}}\approx 0.160511021714$

Galois group

$D_8:C_2$ (as 16T47):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), 4.0.117.1 x2, 4.2.507.1 x2, \(\Q(\sqrt{-3}, \sqrt{13})\), 8.0.2313441.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: Deg 32
Degree 16 siblings: 16.4.232292068597265625.1, 16.0.12370583534765625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
5.8.4.2$x^{8} + 25 x^{4} - 250 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$