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Properties

Label 16.0.5605632412826517.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.606\times 10^{15}$
Root discriminant $9.64$
Ramified primes $3, 37$
Class number $1$
Class group trivial
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 75*x^11 + 104*x^10 - 121*x^9 + 121*x^8 - 107*x^7 + 86*x^6 - 63*x^5 + 43*x^4 - 27*x^3 + 14*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 75*x^11 + 104*x^10 - 121*x^9 + 121*x^8 - 107*x^7 + 86*x^6 - 63*x^5 + 43*x^4 - 27*x^3 + 14*x^2 - 5*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 14, -27, 43, -63, 86, -107, 121, -121, 104, -75, 46, -24, 11, -4, 1]);
 

\( x^{16} - 4 x^{15} + 11 x^{14} - 24 x^{13} + 46 x^{12} - 75 x^{11} + 104 x^{10} - 121 x^{9} + 121 x^{8} - 107 x^{7} + 86 x^{6} - 63 x^{5} + 43 x^{4} - 27 x^{3} + 14 x^{2} - 5 x + 1 \)

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:  \(5605632412826517\)\(\medspace = 3^{10}\cdot 37^{7}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.64$
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:  $3, 37$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{2}{13} a^{13} + \frac{6}{13} a^{12} + \frac{4}{13} a^{11} - \frac{6}{13} a^{10} + \frac{3}{13} a^{9} + \frac{3}{13} a^{8} + \frac{2}{13} a^{7} + \frac{4}{13} a^{6} + \frac{5}{13} a^{4} + \frac{6}{13} a^{3} - \frac{6}{13} a^{2} + \frac{4}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{15} + \frac{2}{13} a^{13} + \frac{5}{13} a^{12} - \frac{1}{13} a^{11} + \frac{2}{13} a^{10} - \frac{3}{13} a^{9} - \frac{4}{13} a^{8} + \frac{5}{13} a^{6} + \frac{5}{13} a^{5} - \frac{4}{13} a^{4} - \frac{5}{13} a^{3} + \frac{3}{13} a^{2} + \frac{2}{13} a + \frac{6}{13}$

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:  \( \frac{14}{13} a^{15} - \frac{56}{13} a^{14} + \frac{150}{13} a^{13} - \frac{331}{13} a^{12} + \frac{620}{13} a^{11} - 78 a^{10} + \frac{1376}{13} a^{9} - \frac{1589}{13} a^{8} + \frac{1539}{13} a^{7} - \frac{1337}{13} a^{6} + \frac{1032}{13} a^{5} - \frac{752}{13} a^{4} + \frac{491}{13} a^{3} - \frac{298}{13} a^{2} + \frac{142}{13} a - \frac{34}{13} \) (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:  \( 32.1900879299 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 32.1900879299 \cdot 1}{6\sqrt{5605632412826517}}\approx 0.174059427860$

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.333.1, 8.0.4102893.1

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 $16$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $16$

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$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$37$37.4.3.4$x^{4} + 296$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$