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Properties

Label 16.0.59734284026...5129.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 79^{8}$
Root discriminant $83.85$
Ramified primes $13, 79$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48841, 0, 671437, 0, 1114217, 0, 166803, 0, 26429, 0, -7683, 0, -130, 0, 52, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 52*x^14 - 130*x^12 - 7683*x^10 + 26429*x^8 + 166803*x^6 + 1114217*x^4 + 671437*x^2 + 48841)
 
gp: K = bnfinit(x^16 + 52*x^14 - 130*x^12 - 7683*x^10 + 26429*x^8 + 166803*x^6 + 1114217*x^4 + 671437*x^2 + 48841, 1)
 

Normalized defining polynomial

\( x^{16} + 52 x^{14} - 130 x^{12} - 7683 x^{10} + 26429 x^{8} + 166803 x^{6} + 1114217 x^{4} + 671437 x^{2} + 48841 \)

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:  \(5973428402662444840532734135129=13^{14}\cdot 79^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.85$
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:  $13, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{78} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{78} a^{9} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{78} a^{10} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{1326} a^{11} + \frac{2}{663} a^{9} - \frac{3}{34} a^{7} - \frac{1}{6} a^{6} - \frac{4}{17} a^{5} + \frac{1}{3} a^{4} + \frac{7}{51} a^{3} - \frac{1}{6} a^{2} - \frac{37}{102} a - \frac{1}{6}$, $\frac{1}{11934} a^{12} + \frac{19}{5967} a^{10} + \frac{2}{663} a^{8} - \frac{1}{6} a^{7} + \frac{5}{459} a^{6} - \frac{1}{6} a^{5} - \frac{137}{306} a^{4} + \frac{1}{3} a^{3} + \frac{7}{459} a^{2} + \frac{1}{3} a + \frac{17}{54}$, $\frac{1}{11934} a^{13} + \frac{1}{5967} a^{11} + \frac{5}{1326} a^{9} + \frac{14}{459} a^{7} + \frac{49}{306} a^{5} + \frac{61}{459} a^{3} - \frac{215}{918} a - \frac{1}{2}$, $\frac{1}{16066014338148606} a^{14} - \frac{9037259252}{2677669056358101} a^{12} + \frac{336954076435}{617923628390331} a^{10} + \frac{50901870886897}{16066014338148606} a^{8} - \frac{1}{6} a^{7} - \frac{92593291132558}{617923628390331} a^{6} + \frac{1}{3} a^{5} + \frac{73096504190167}{617923628390331} a^{4} + \frac{1}{3} a^{3} - \frac{8886711260099}{411949085593554} a^{2} + \frac{1}{3} a - \frac{2317011083113}{72696897457686}$, $\frac{1}{208858186395931878} a^{15} - \frac{35909292239}{5355338112716202} a^{13} - \frac{6057286056175}{16066014338148606} a^{11} - \frac{10996655564123}{16066014338148606} a^{9} - \frac{1931258360091389}{16066014338148606} a^{7} + \frac{3384442401217}{72696897457686} a^{5} + \frac{186478124782259}{411949085593554} a^{3} - \frac{1}{2} a^{2} - \frac{87946541438177}{1235847256780662} a - \frac{1}{2}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  $7$
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:  \( 79431585.07 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

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 $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{-1027}) \), \(\Q(\sqrt{-79}) \), 4.4.13711477.1, 4.0.2197.1, \(\Q(\sqrt{13}, \sqrt{-79})\), 8.4.2444059819779877.1 x2, 8.0.391613494597.1 x2, 8.0.188004601521529.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$79$79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$