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Properties

Label 16.8.23933974895...4736.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{52}\cdot 3^{12}$
Root discriminant $21.69$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2:Q_8$ (as 16T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -80, 160, 512, -1744, 1240, 1328, -2680, 1378, 368, -820, 424, -64, -40, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 40*x^13 - 64*x^12 + 424*x^11 - 820*x^10 + 368*x^9 + 1378*x^8 - 2680*x^7 + 1328*x^6 + 1240*x^5 - 1744*x^4 + 512*x^3 + 160*x^2 - 80*x - 2)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 40*x^13 - 64*x^12 + 424*x^11 - 820*x^10 + 368*x^9 + 1378*x^8 - 2680*x^7 + 1328*x^6 + 1240*x^5 - 1744*x^4 + 512*x^3 + 160*x^2 - 80*x - 2, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 40 x^{13} - 64 x^{12} + 424 x^{11} - 820 x^{10} + 368 x^{9} + 1378 x^{8} - 2680 x^{7} + 1328 x^{6} + 1240 x^{5} - 1744 x^{4} + 512 x^{3} + 160 x^{2} - 80 x - 2 \)

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:  \(2393397489569403764736=2^{52}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.69$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{9407} a^{14} + \frac{3389}{9407} a^{13} + \frac{1001}{9407} a^{12} + \frac{610}{9407} a^{11} + \frac{4058}{9407} a^{10} + \frac{443}{9407} a^{9} + \frac{619}{9407} a^{8} + \frac{898}{9407} a^{7} - \frac{3585}{9407} a^{6} - \frac{3489}{9407} a^{5} + \frac{1859}{9407} a^{4} + \frac{582}{9407} a^{3} + \frac{1267}{9407} a^{2} + \frac{1192}{9407} a + \frac{1855}{9407}$, $\frac{1}{1732219193977} a^{15} - \frac{17290909}{1732219193977} a^{14} - \frac{458036025618}{1732219193977} a^{13} - \frac{410971090037}{1732219193977} a^{12} + \frac{101923744437}{1732219193977} a^{11} + \frac{696229815771}{1732219193977} a^{10} - \frac{44561139897}{1732219193977} a^{9} - \frac{93435661333}{1732219193977} a^{8} - \frac{481724110670}{1732219193977} a^{7} - \frac{212323803070}{1732219193977} a^{6} - \frac{204782884897}{1732219193977} a^{5} + \frac{1049152722}{4235254753} a^{4} - \frac{320655465423}{1732219193977} a^{3} + \frac{190720473036}{1732219193977} a^{2} + \frac{623535403014}{1732219193977} a + \frac{3740376361}{75313877999}$

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:  \( 90901.8794542 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:Q_8$ (as 16T31):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:Q_8$
Character table for $C_2^2:Q_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.2.55296.4, \(\Q(\sqrt{2}, \sqrt{3})\), 4.2.55296.1, 8.4.12230590464.5, 8.4.84934656.1, 8.8.12230590464.1

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 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed