↑

Properties

Label 16.0.7058653305387264.1
Degree $16$
Signature $[0, 8]$
Discriminant $7.059\times 10^{15}$
Root discriminant $9.78$
Ramified primes $2, 3, 7$
Class number $1$
Class group trivial
Galois group $D_8:C_2$ (as 16T47)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 3*x^13 + 11*x^12 - 19*x^11 + 10*x^10 + 21*x^9 - 44*x^8 + 21*x^7 + 40*x^6 - 89*x^5 + 92*x^4 - 60*x^3 + 26*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 - 3*x^13 + 11*x^12 - 19*x^11 + 10*x^10 + 21*x^9 - 44*x^8 + 21*x^7 + 40*x^6 - 89*x^5 + 92*x^4 - 60*x^3 + 26*x^2 - 7*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 26, -60, 92, -89, 40, 21, -44, 21, 10, -19, 11, -3, 2, -2, 1]);
 

\( x^{16} - 2 x^{15} + 2 x^{14} - 3 x^{13} + 11 x^{12} - 19 x^{11} + 10 x^{10} + 21 x^{9} - 44 x^{8} + 21 x^{7} + 40 x^{6} - 89 x^{5} + 92 x^{4} - 60 x^{3} + 26 x^{2} - 7 x + 1 \)

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:  \(7058653305387264\)\(\medspace = 2^{8}\cdot 3^{14}\cdot 7^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.78$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{40937} a^{15} - \frac{197}{3149} a^{14} + \frac{3681}{40937} a^{13} - \frac{4172}{40937} a^{12} - \frac{646}{3149} a^{11} - \frac{1462}{40937} a^{10} + \frac{16001}{40937} a^{9} - \frac{9538}{40937} a^{8} + \frac{138}{611} a^{7} + \frac{1093}{40937} a^{6} - \frac{13231}{40937} a^{5} + \frac{3141}{40937} a^{4} - \frac{14075}{40937} a^{3} - \frac{515}{3149} a^{2} - \frac{1544}{3149} a - \frac{11694}{40937}$

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:  \( -\frac{324}{3149} a^{15} - \frac{4721}{3149} a^{14} + \frac{3976}{3149} a^{13} + \frac{807}{3149} a^{12} + \frac{6514}{3149} a^{11} - \frac{39599}{3149} a^{10} + \frac{36718}{3149} a^{9} + \frac{32633}{3149} a^{8} - \frac{1566}{47} a^{7} + \frac{67834}{3149} a^{6} + \frac{86078}{3149} a^{5} - \frac{180050}{3149} a^{4} + \frac{139104}{3149} a^{3} - \frac{54014}{3149} a^{2} + \frac{3792}{3149} a + \frac{3758}{3149} \) (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:  \( 37.0149239843 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 37.0149239843 \cdot 1}{6\sqrt{7058653305387264}}\approx 0.178362533511$

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

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), 4.2.1323.1 x2, 4.0.189.1 x2, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.1750329.1

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: 16.4.1807015246179139584.1, 16.0.36877862166921216.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
3Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$