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Properties

Label 16.8.74796181841...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{12}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $20.17$
Ramified primes $3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 14, -39, 89, -84, 53, -42, -77, 141, -98, 63, -16, -3, 1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + x^14 - 3*x^13 - 16*x^12 + 63*x^11 - 98*x^10 + 141*x^9 - 77*x^8 - 42*x^7 + 53*x^6 - 84*x^5 + 89*x^4 - 39*x^3 + 14*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 3*x^15 + x^14 - 3*x^13 - 16*x^12 + 63*x^11 - 98*x^10 + 141*x^9 - 77*x^8 - 42*x^7 + 53*x^6 - 84*x^5 + 89*x^4 - 39*x^3 + 14*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + x^{14} - 3 x^{13} - 16 x^{12} + 63 x^{11} - 98 x^{10} + 141 x^{9} - 77 x^{8} - 42 x^{7} + 53 x^{6} - 84 x^{5} + 89 x^{4} - 39 x^{3} + 14 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(747961818418212890625=3^{12}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.17$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{16444} a^{14} + \frac{140}{4111} a^{13} + \frac{393}{8222} a^{12} - \frac{3433}{16444} a^{11} - \frac{844}{4111} a^{10} - \frac{7827}{16444} a^{9} + \frac{3999}{8222} a^{8} + \frac{3127}{8222} a^{7} + \frac{2693}{16444} a^{6} + \frac{3023}{8222} a^{5} + \frac{895}{16444} a^{4} + \frac{1441}{4111} a^{3} + \frac{1169}{4111} a^{2} - \frac{2221}{16444} a + \frac{881}{4111}$, $\frac{1}{1463516} a^{15} - \frac{9}{1463516} a^{14} + \frac{33589}{365879} a^{13} + \frac{32246}{365879} a^{12} + \frac{36318}{365879} a^{11} - \frac{46901}{365879} a^{10} - \frac{414085}{1463516} a^{9} + \frac{162924}{365879} a^{8} - \frac{174672}{365879} a^{7} - \frac{84548}{365879} a^{6} + \frac{4518}{365879} a^{5} - \frac{10171}{1463516} a^{4} - \frac{119890}{365879} a^{3} - \frac{252295}{731758} a^{2} - \frac{122031}{365879} a + \frac{276449}{1463516}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 38476.3535658 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{15})^+\), 4.4.6125.1, 4.2.1323.1, \(\Q(\sqrt{5}, \sqrt{21})\), 4.2.33075.1, 8.4.1093955625.1, 8.8.3038765625.1, 8.4.27348890625.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
3Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$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}$