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Properties

Label 16.0.12357173778...1369.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{12}\cdot 13^{14}$
Root discriminant $56.98$
Ramified primes $11, 13$
Class number $125$ (GRH)
Class group $[5, 5, 5]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![531441, 0, 702027, 0, 169128, 0, -11421, 0, -3609, 0, -292, 0, -54, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 12*x^14 - 54*x^12 - 292*x^10 - 3609*x^8 - 11421*x^6 + 169128*x^4 + 702027*x^2 + 531441)
 
gp: K = bnfinit(x^16 + 12*x^14 - 54*x^12 - 292*x^10 - 3609*x^8 - 11421*x^6 + 169128*x^4 + 702027*x^2 + 531441, 1)
 

Normalized defining polynomial

\( x^{16} + 12 x^{14} - 54 x^{12} - 292 x^{10} - 3609 x^{8} - 11421 x^{6} + 169128 x^{4} + 702027 x^{2} + 531441 \)

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:  \(12357173778709817574713851369=11^{12}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.98$
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:  $11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{7} - \frac{2}{27} a^{5} + \frac{1}{27} a^{3} - \frac{1}{3} a$, $\frac{1}{54} a^{8} - \frac{1}{27} a^{6} + \frac{1}{54} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{162} a^{9} - \frac{1}{54} a^{5} - \frac{1}{6} a^{4} + \frac{10}{81} a^{3} + \frac{1}{6} a^{2} - \frac{5}{18} a$, $\frac{1}{486} a^{10} + \frac{1}{162} a^{8} - \frac{1}{54} a^{6} - \frac{1}{6} a^{5} - \frac{31}{486} a^{4} + \frac{7}{54} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{1458} a^{11} + \frac{1}{486} a^{9} - \frac{1}{54} a^{7} - \frac{1}{18} a^{6} + \frac{167}{1458} a^{5} + \frac{1}{9} a^{4} - \frac{13}{162} a^{3} - \frac{1}{18} a^{2} - \frac{1}{18} a$, $\frac{1}{39366} a^{12} - \frac{4}{6561} a^{10} - \frac{1}{162} a^{8} - \frac{1}{54} a^{7} + \frac{826}{19683} a^{6} - \frac{7}{54} a^{5} + \frac{119}{4374} a^{4} + \frac{4}{27} a^{3} - \frac{26}{243} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{118098} a^{13} - \frac{4}{19683} a^{11} - \frac{1}{486} a^{9} - \frac{632}{59049} a^{7} - \frac{1}{18} a^{6} - \frac{2149}{13122} a^{5} - \frac{1}{18} a^{4} - \frac{7}{1458} a^{3} + \frac{1}{9} a^{2} + \frac{5}{18} a - \frac{1}{2}$, $\frac{1}{9924129234} a^{14} + \frac{24577}{3308043078} a^{12} + \frac{135203}{367560342} a^{10} - \frac{66881911}{9924129234} a^{8} - \frac{270169}{13613346} a^{6} - \frac{310960}{20420019} a^{4} - \frac{2467439}{13613346} a^{2} - \frac{1}{2} a - \frac{28895}{84033}$, $\frac{1}{29772387702} a^{15} + \frac{24577}{9924129234} a^{13} + \frac{135203}{1102681026} a^{11} - \frac{66881911}{29772387702} a^{9} + \frac{234029}{40840038} a^{7} - \frac{1823554}{61260057} a^{5} - \frac{1963241}{40840038} a^{3} - \frac{1}{2} a^{2} + \frac{111160}{252099} a$

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

Class group and class number

$C_{5}\times C_{5}\times C_{5}$, which has order $125$ (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:  \( 61187181.075 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-143}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{13})\), 4.4.265837.1, 4.0.2197.1, 8.0.70669310569.1, 8.4.111162825525037.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
11Data not computed
13Data not computed