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Properties

Label 16.0.6561000000000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $6.561\times 10^{15}$
Root discriminant $9.74$
Ramified primes $2, 3, 5$
Class number $1$
Class group trivial
Galois group $C_4 \times D_4$ (as 16T19)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 15*x^14 - 32*x^13 + 56*x^12 - 85*x^11 + 114*x^10 - 134*x^9 + 131*x^8 - 98*x^7 + 42*x^6 + 9*x^5 - 28*x^4 + 18*x^3 - x^2 - 3*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 15*x^14 - 32*x^13 + 56*x^12 - 85*x^11 + 114*x^10 - 134*x^9 + 131*x^8 - 98*x^7 + 42*x^6 + 9*x^5 - 28*x^4 + 18*x^3 - x^2 - 3*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -1, 18, -28, 9, 42, -98, 131, -134, 114, -85, 56, -32, 15, -5, 1]);
 

\( x^{16} - 5 x^{15} + 15 x^{14} - 32 x^{13} + 56 x^{12} - 85 x^{11} + 114 x^{10} - 134 x^{9} + 131 x^{8} - 98 x^{7} + 42 x^{6} + 9 x^{5} - 28 x^{4} + 18 x^{3} - x^{2} - 3 x + 1 \)

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:  \(6561000000000000\)\(\medspace = 2^{12}\cdot 3^{8}\cdot 5^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.74$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13129} a^{15} - \frac{1370}{13129} a^{14} + \frac{5747}{13129} a^{13} + \frac{6455}{13129} a^{12} - \frac{1460}{13129} a^{11} - \frac{147}{691} a^{10} + \frac{271}{691} a^{9} - \frac{4504}{13129} a^{8} + \frac{3719}{13129} a^{7} + \frac{4390}{13129} a^{6} - \frac{5484}{13129} a^{5} + \frac{2139}{13129} a^{4} - \frac{5125}{13129} a^{3} - \frac{2114}{13129} a^{2} - \frac{2771}{13129} a + \frac{1260}{13129}$

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:  \( \frac{4711}{13129} a^{15} - \frac{20860}{13129} a^{14} + \frac{67764}{13129} a^{13} - \frac{154807}{13129} a^{12} + \frac{290374}{13129} a^{11} - \frac{24320}{691} a^{10} + \frac{34263}{691} a^{9} - \frac{815878}{13129} a^{8} + \frac{872637}{13129} a^{7} - \frac{771496}{13129} a^{6} + \frac{501650}{13129} a^{5} - \frac{163791}{13129} a^{4} - \frac{65289}{13129} a^{3} + \frac{110889}{13129} a^{2} - \frac{30213}{13129} a - \frac{11577}{13129} \) (order $30$)
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:  \( 175.01473438 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 175.01473438 \cdot 1}{30\sqrt{6561000000000000}}\approx 0.17494731698$

Galois group

$C_4\times D_4$ (as 16T19):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.9000.2, 4.0.9000.1, 8.0.81000000.1, \(\Q(\zeta_{15})\), 8.0.3240000.1

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: 16.0.331776000000000000.1, 16.8.26873856000000000000.4, 16.0.26873856000000000000.11

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$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}$
5Data not computed