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Properties

Label 16.8.67108864000...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{40}\cdot 5^{14}$
Root discriminant $23.13$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2 \times (C_8:C_2)$ (as 16T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, -188, 184, 412, -1022, 492, 636, -924, 538, -268, 104, 36, -82, 44, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 44*x^13 - 82*x^12 + 36*x^11 + 104*x^10 - 268*x^9 + 538*x^8 - 924*x^7 + 636*x^6 + 492*x^5 - 1022*x^4 + 412*x^3 + 184*x^2 - 188*x + 41)
 
gp: K = bnfinit(x^16 - 4*x^15 - 4*x^14 + 44*x^13 - 82*x^12 + 36*x^11 + 104*x^10 - 268*x^9 + 538*x^8 - 924*x^7 + 636*x^6 + 492*x^5 - 1022*x^4 + 412*x^3 + 184*x^2 - 188*x + 41, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 4 x^{14} + 44 x^{13} - 82 x^{12} + 36 x^{11} + 104 x^{10} - 268 x^{9} + 538 x^{8} - 924 x^{7} + 636 x^{6} + 492 x^{5} - 1022 x^{4} + 412 x^{3} + 184 x^{2} - 188 x + 41 \)

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:  \(6710886400000000000000=2^{40}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.13$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{82} a^{13} - \frac{15}{82} a^{12} + \frac{5}{82} a^{11} + \frac{3}{82} a^{10} + \frac{3}{41} a^{9} + \frac{4}{41} a^{8} - \frac{13}{41} a^{7} - \frac{8}{41} a^{6} - \frac{21}{82} a^{5} - \frac{1}{2} a^{4} + \frac{5}{82} a^{3} + \frac{1}{82} a^{2} + \frac{16}{41} a$, $\frac{1}{1558} a^{14} + \frac{4}{779} a^{13} + \frac{275}{1558} a^{12} + \frac{77}{1558} a^{11} - \frac{9}{82} a^{10} + \frac{105}{1558} a^{9} + \frac{281}{1558} a^{8} - \frac{307}{779} a^{7} + \frac{27}{82} a^{6} + \frac{148}{779} a^{5} - \frac{733}{1558} a^{4} + \frac{321}{1558} a^{3} - \frac{23}{82} a^{2} - \frac{125}{1558} a + \frac{11}{38}$, $\frac{1}{520311238} a^{15} - \frac{44259}{260155619} a^{14} + \frac{178594}{260155619} a^{13} - \frac{73188177}{520311238} a^{12} - \frac{21632861}{260155619} a^{11} + \frac{120384751}{520311238} a^{10} + \frac{23973280}{260155619} a^{9} - \frac{25459062}{260155619} a^{8} + \frac{5169721}{520311238} a^{7} - \frac{54765224}{260155619} a^{6} + \frac{9388627}{260155619} a^{5} - \frac{119326483}{520311238} a^{4} - \frac{37740493}{260155619} a^{3} - \frac{253058883}{520311238} a^{2} - \frac{116829740}{260155619} a + \frac{3159758}{6345259}$

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:  \( 102168.613642 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times OD_{16}$ (as 16T15):

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 $C_2 \times (C_8:C_2)$
Character table for $C_2 \times (C_8:C_2)$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{20})^+\), 4.4.8000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.5120000000.2, 8.4.5120000000.1, \(\Q(\zeta_{40})^+\)

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 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
5Data not computed