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Properties

Label 16.0.46116860184...7904.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}$
Root discriminant $14.67$
Ramified prime $2$
Class number $1$
Class group Trivial
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -8, 24, -104, 312, -696, 1192, -1624, 1800, -1624, 1208, -728, 356, -136, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 - 136*x^13 + 356*x^12 - 728*x^11 + 1208*x^10 - 1624*x^9 + 1800*x^8 - 1624*x^7 + 1192*x^6 - 696*x^5 + 312*x^4 - 104*x^3 + 24*x^2 - 8*x + 5)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 - 136*x^13 + 356*x^12 - 728*x^11 + 1208*x^10 - 1624*x^9 + 1800*x^8 - 1624*x^7 + 1192*x^6 - 696*x^5 + 312*x^4 - 104*x^3 + 24*x^2 - 8*x + 5, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} - 136 x^{13} + 356 x^{12} - 728 x^{11} + 1208 x^{10} - 1624 x^{9} + 1800 x^{8} - 1624 x^{7} + 1192 x^{6} - 696 x^{5} + 312 x^{4} - 104 x^{3} + 24 x^{2} - 8 x + 5 \)

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:  \(4611686018427387904=2^{62}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.67$
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:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} - \frac{10}{31} a^{13} + \frac{4}{31} a^{12} + \frac{13}{31} a^{11} + \frac{13}{31} a^{10} + \frac{6}{31} a^{9} + \frac{3}{31} a^{8} - \frac{13}{31} a^{7} + \frac{15}{31} a^{6} + \frac{4}{31} a^{5} + \frac{3}{31} a^{4} + \frac{4}{31} a^{3} + \frac{12}{31} a^{2} - \frac{11}{31} a - \frac{6}{31}$, $\frac{1}{397122679} a^{15} - \frac{4976387}{397122679} a^{14} + \frac{60898044}{397122679} a^{13} + \frac{63131353}{397122679} a^{12} - \frac{20117089}{397122679} a^{11} - \frac{12596837}{397122679} a^{10} + \frac{13178514}{397122679} a^{9} - \frac{172093781}{397122679} a^{8} + \frac{19786316}{397122679} a^{7} - \frac{9099003}{397122679} a^{6} - \frac{160668476}{397122679} a^{5} + \frac{66386105}{397122679} a^{4} - \frac{97432218}{397122679} a^{3} + \frac{36207416}{397122679} a^{2} + \frac{772851}{12810409} a - \frac{122419039}{397122679}$

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:  \( \frac{1260092}{9685919} a^{15} - \frac{8992308}{9685919} a^{14} + \frac{42348236}{9685919} a^{13} - \frac{133470344}{9685919} a^{12} + \frac{329339492}{9685919} a^{11} - \frac{629351036}{9685919} a^{10} + \frac{989216244}{9685919} a^{9} - \frac{1253769647}{9685919} a^{8} + \frac{1336693868}{9685919} a^{7} - \frac{1149688844}{9685919} a^{6} + \frac{820970636}{9685919} a^{5} - \frac{458895758}{9685919} a^{4} + \frac{205357172}{9685919} a^{3} - \frac{68834116}{9685919} a^{2} + \frac{790348}{312449} a - \frac{9034238}{9685919} \) (order $4$)
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:  \( 2661.0561913 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.512.1, 8.0.134217728.2, 8.0.134217728.1, 8.0.67108864.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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