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Properties

Label 16.0.34789235097...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{4}\cdot 5^{8}$
Root discriminant $16.65$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_2 \times (C_4\times C_2):C_2$ (as 16T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -16, 32, 0, -44, 24, 112, 0, -126, 0, 200, 192, 58, -16, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 - 16*x^13 + 58*x^12 + 192*x^11 + 200*x^10 - 126*x^8 + 112*x^6 + 24*x^5 - 44*x^4 + 32*x^2 - 16*x + 4)
 
gp: K = bnfinit(x^16 - 12*x^14 - 16*x^13 + 58*x^12 + 192*x^11 + 200*x^10 - 126*x^8 + 112*x^6 + 24*x^5 - 44*x^4 + 32*x^2 - 16*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{14} - 16 x^{13} + 58 x^{12} + 192 x^{11} + 200 x^{10} - 126 x^{8} + 112 x^{6} + 24 x^{5} - 44 x^{4} + 32 x^{2} - 16 x + 4 \)

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:  \(34789235097600000000=2^{40}\cdot 3^{4}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.65$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{1187775155906} a^{15} + \frac{119475791645}{593887577953} a^{14} - \frac{90312007974}{593887577953} a^{13} + \frac{73274592611}{1187775155906} a^{12} + \frac{208992905625}{1187775155906} a^{11} + \frac{6459726878}{34934563409} a^{10} + \frac{252240318681}{1187775155906} a^{9} + \frac{156549044243}{1187775155906} a^{8} + \frac{131236830640}{593887577953} a^{7} + \frac{251957848277}{593887577953} a^{6} + \frac{234391366198}{593887577953} a^{5} - \frac{97257322605}{593887577953} a^{4} - \frac{28094613401}{593887577953} a^{3} + \frac{197363833407}{593887577953} a^{2} + \frac{76665966453}{593887577953} a + \frac{108430013145}{593887577953}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{24901223125}{69869126818} a^{15} + \frac{14419726409}{69869126818} a^{14} - \frac{297169312493}{69869126818} a^{13} - \frac{285726227051}{34934563409} a^{12} + \frac{595692928422}{34934563409} a^{11} + \frac{5589742594753}{69869126818} a^{10} + \frac{3936399610029}{34934563409} a^{9} + \frac{1627719915466}{34934563409} a^{8} - \frac{1462671630840}{34934563409} a^{7} - \frac{1179861622766}{34934563409} a^{6} + \frac{920920909789}{34934563409} a^{5} + \frac{967234592108}{34934563409} a^{4} - \frac{158935429982}{34934563409} a^{3} - \frac{231646848555}{34934563409} a^{2} + \frac{263154246082}{34934563409} a - \frac{25893886559}{34934563409} \) (order $8$)
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:  \( 5307.66048935 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:C_2$ (as 16T18):

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 $C_2 \times (C_4\times C_2):C_2$
Character table for $C_2 \times (C_4\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\zeta_{8})\), 8.0.40960000.1, 8.0.9437184.1, 8.0.5898240000.5

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$