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Properties

Label 16.0.11957422500...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 3^{14}\cdot 5^{14}$
Root discriminant $17.98$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $C_8:C_2^2$ (as 16T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -384, 416, -336, 148, -384, 554, -408, 193, 111, -109, 12, -2, 12, -1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - x^14 + 12*x^13 - 2*x^12 + 12*x^11 - 109*x^10 + 111*x^9 + 193*x^8 - 408*x^7 + 554*x^6 - 384*x^5 + 148*x^4 - 336*x^3 + 416*x^2 - 384*x + 256)
 
gp: K = bnfinit(x^16 - 3*x^15 - x^14 + 12*x^13 - 2*x^12 + 12*x^11 - 109*x^10 + 111*x^9 + 193*x^8 - 408*x^7 + 554*x^6 - 384*x^5 + 148*x^4 - 336*x^3 + 416*x^2 - 384*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - x^{14} + 12 x^{13} - 2 x^{12} + 12 x^{11} - 109 x^{10} + 111 x^{9} + 193 x^{8} - 408 x^{7} + 554 x^{6} - 384 x^{5} + 148 x^{4} - 336 x^{3} + 416 x^{2} - 384 x + 256 \)

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:  \(119574225000000000000=2^{12}\cdot 3^{14}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.98$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{8} + \frac{3}{16} a^{7} + \frac{3}{32} a^{6} + \frac{3}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2}$, $\frac{1}{352} a^{13} - \frac{1}{176} a^{12} - \frac{1}{11} a^{10} - \frac{1}{176} a^{9} + \frac{1}{176} a^{8} + \frac{47}{352} a^{7} + \frac{2}{11} a^{6} + \frac{29}{176} a^{5} + \frac{5}{22} a^{4} - \frac{43}{88} a^{3} - \frac{7}{44} a^{2} - \frac{1}{11} a - \frac{5}{11}$, $\frac{1}{20768} a^{14} + \frac{3}{2596} a^{13} + \frac{21}{2596} a^{12} - \frac{85}{5192} a^{11} - \frac{813}{10384} a^{10} + \frac{1075}{10384} a^{9} + \frac{209}{1888} a^{8} - \frac{2393}{10384} a^{7} + \frac{1543}{10384} a^{6} - \frac{217}{1298} a^{5} - \frac{2207}{5192} a^{4} + \frac{1073}{2596} a^{3} - \frac{173}{649} a^{2} - \frac{227}{1298} a - \frac{218}{649}$, $\frac{1}{14490373568} a^{15} - \frac{41759}{14490373568} a^{14} + \frac{15421501}{14490373568} a^{13} - \frac{43034557}{3622593392} a^{12} + \frac{347888303}{7245186784} a^{11} - \frac{177374951}{3622593392} a^{10} - \frac{994109873}{14490373568} a^{9} - \frac{1392344289}{14490373568} a^{8} - \frac{2044742845}{14490373568} a^{7} - \frac{352113891}{3622593392} a^{6} + \frac{80670455}{7245186784} a^{5} + \frac{654587443}{1811296696} a^{4} - \frac{56227947}{329326672} a^{3} - \frac{134801863}{905648348} a^{2} - \frac{10180359}{41165834} a - \frac{62542223}{226412087}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{105009}{245599552} a^{15} + \frac{834805}{245599552} a^{14} - \frac{4017985}{245599552} a^{13} - \frac{522685}{122799776} a^{12} + \frac{10028735}{122799776} a^{11} + \frac{283311}{61399888} a^{10} - \frac{25359645}{245599552} a^{9} - \frac{138139165}{245599552} a^{8} + \frac{175616445}{245599552} a^{7} + \frac{199397515}{122799776} a^{6} - \frac{301069933}{122799776} a^{5} + \frac{17360895}{61399888} a^{4} - \frac{10334505}{61399888} a^{3} - \frac{15520255}{30699944} a^{2} - \frac{752345}{697726} a + \frac{9808110}{3837493} \) (order $6$)
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:  \( 5726.12322965 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.0.13500.2, 4.0.13500.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.10935000000.2, 8.0.10935000000.1, 8.0.182250000.1

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 8 siblings: 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
3Data not computed
5Data not computed