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Properties

Label 16.0.57355639689...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 59^{8}$
Root discriminant $17.18$
Ramified primes $5, 59$
Class number $3$
Class group $[3]$
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![441, 546, 527, 206, 574, -349, 649, -413, 402, -155, 137, -43, 27, -7, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^14 - 7*x^13 + 27*x^12 - 43*x^11 + 137*x^10 - 155*x^9 + 402*x^8 - 413*x^7 + 649*x^6 - 349*x^5 + 574*x^4 + 206*x^3 + 527*x^2 + 546*x + 441)
 
gp: K = bnfinit(x^16 + 6*x^14 - 7*x^13 + 27*x^12 - 43*x^11 + 137*x^10 - 155*x^9 + 402*x^8 - 413*x^7 + 649*x^6 - 349*x^5 + 574*x^4 + 206*x^3 + 527*x^2 + 546*x + 441, 1)
 

Normalized defining polynomial

\( x^{16} + 6 x^{14} - 7 x^{13} + 27 x^{12} - 43 x^{11} + 137 x^{10} - 155 x^{9} + 402 x^{8} - 413 x^{7} + 649 x^{6} - 349 x^{5} + 574 x^{4} + 206 x^{3} + 527 x^{2} + 546 x + 441 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57355639689187890625=5^{8}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.18$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{42} a^{12} + \frac{5}{42} a^{11} - \frac{1}{7} a^{10} + \frac{5}{42} a^{9} + \frac{1}{21} a^{8} - \frac{19}{42} a^{7} + \frac{11}{42} a^{6} + \frac{1}{3} a^{5} - \frac{10}{21} a^{4} + \frac{1}{14} a^{3} + \frac{11}{42} a^{2} - \frac{1}{14} a - \frac{1}{2}$, $\frac{1}{42} a^{13} - \frac{1}{14} a^{11} - \frac{1}{6} a^{10} + \frac{5}{42} a^{9} - \frac{1}{42} a^{8} + \frac{4}{21} a^{7} - \frac{13}{42} a^{6} - \frac{10}{21} a^{5} + \frac{5}{42} a^{4} - \frac{2}{21} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a - \frac{1}{2}$, $\frac{1}{126} a^{14} - \frac{1}{126} a^{13} - \frac{1}{126} a^{12} + \frac{10}{63} a^{11} + \frac{2}{63} a^{9} + \frac{13}{126} a^{8} + \frac{25}{126} a^{7} + \frac{5}{42} a^{6} - \frac{31}{126} a^{5} - \frac{7}{18} a^{4} + \frac{25}{63} a^{3} - \frac{19}{63} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4927530202761474} a^{15} + \frac{177178330771}{351966443054391} a^{14} - \frac{28832325788960}{2463765101380737} a^{13} + \frac{49839273716873}{4927530202761474} a^{12} - \frac{10113273177824}{273751677931193} a^{11} + \frac{10723912286855}{2463765101380737} a^{10} - \frac{374728989542861}{4927530202761474} a^{9} - \frac{89571611115808}{2463765101380737} a^{8} - \frac{146366450767057}{821255033793579} a^{7} - \frac{480812801808986}{2463765101380737} a^{6} - \frac{203872515868922}{2463765101380737} a^{5} - \frac{283739461374667}{703932886108782} a^{4} - \frac{110549309456914}{2463765101380737} a^{3} + \frac{38302983713735}{78214765123198} a^{2} + \frac{146744650733216}{821255033793579} a - \frac{38928104432583}{78214765123198}$

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

Class group and class number

$C_{3}$, which has order $3$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 859.305557763 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-295}) \), \(\Q(\sqrt{-59}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-59})\), 4.0.17405.1 x2, 4.2.1475.1 x2, 8.0.7573350625.1, 8.0.1514670125.1 x4, 8.2.128361875.1 x4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ R

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$59$59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$