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Properties

Label 16.0.11819246862...024.14
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 3^{8}$
Root discriminant $17.97$
Ramified primes $2, 3$
Class number $2$
Class group $[2]$
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![46, -496, 2328, -6176, 10284, -11312, 8352, -3912, 686, 624, -680, 320, -44, -40, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 40*x^13 - 44*x^12 + 320*x^11 - 680*x^10 + 624*x^9 + 686*x^8 - 3912*x^7 + 8352*x^6 - 11312*x^5 + 10284*x^4 - 6176*x^3 + 2328*x^2 - 496*x + 46)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 40*x^13 - 44*x^12 + 320*x^11 - 680*x^10 + 624*x^9 + 686*x^8 - 3912*x^7 + 8352*x^6 - 11312*x^5 + 10284*x^4 - 6176*x^3 + 2328*x^2 - 496*x + 46, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 40 x^{13} - 44 x^{12} + 320 x^{11} - 680 x^{10} + 624 x^{9} + 686 x^{8} - 3912 x^{7} + 8352 x^{6} - 11312 x^{5} + 10284 x^{4} - 6176 x^{3} + 2328 x^{2} - 496 x + 46 \)

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:  \(118192468620711297024=2^{54}\cdot 3^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.97$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{97} a^{14} - \frac{44}{97} a^{13} - \frac{34}{97} a^{12} - \frac{15}{97} a^{11} + \frac{6}{97} a^{10} - \frac{38}{97} a^{9} + \frac{27}{97} a^{8} + \frac{23}{97} a^{7} + \frac{36}{97} a^{6} + \frac{2}{97} a^{5} + \frac{46}{97} a^{4} + \frac{36}{97} a^{3} + \frac{8}{97} a^{2} + \frac{46}{97} a + \frac{17}{97}$, $\frac{1}{38097207204359} a^{15} + \frac{47981941174}{38097207204359} a^{14} - \frac{10040909177412}{38097207204359} a^{13} + \frac{1297940936330}{38097207204359} a^{12} + \frac{17160323073826}{38097207204359} a^{11} + \frac{13436676364507}{38097207204359} a^{10} - \frac{2958268494045}{38097207204359} a^{9} + \frac{15376032458693}{38097207204359} a^{8} - \frac{10546791129125}{38097207204359} a^{7} - \frac{9716787033755}{38097207204359} a^{6} - \frac{17884002999267}{38097207204359} a^{5} - \frac{4091063897315}{38097207204359} a^{4} - \frac{3862929481830}{38097207204359} a^{3} - \frac{18886935996764}{38097207204359} a^{2} + \frac{2041789493790}{38097207204359} a + \frac{327787421993}{1656400313233}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 1908.14511711 \)
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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.0.18432.2, 4.0.2048.2, \(\Q(\sqrt{2}, \sqrt{3})\), 4.2.4608.2, 4.2.4608.1, 8.4.339738624.1, 8.4.1358954496.1, 8.0.1358954496.3

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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

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
2Data not computed
$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}$