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Properties

Label 16.16.1919581121...9424.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{32}\cdot 41^{6}\cdot 97^{2}$
Root discriminant $28.52$
Ramified primes $2, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T876

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -36, -198, 2060, -4152, 460, 5792, -3816, -2335, 2628, 164, -720, 86, 88, -18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 18*x^14 + 88*x^13 + 86*x^12 - 720*x^11 + 164*x^10 + 2628*x^9 - 2335*x^8 - 3816*x^7 + 5792*x^6 + 460*x^5 - 4152*x^4 + 2060*x^3 - 198*x^2 - 36*x - 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 18*x^14 + 88*x^13 + 86*x^12 - 720*x^11 + 164*x^10 + 2628*x^9 - 2335*x^8 - 3816*x^7 + 5792*x^6 + 460*x^5 - 4152*x^4 + 2060*x^3 - 198*x^2 - 36*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 18 x^{14} + 88 x^{13} + 86 x^{12} - 720 x^{11} + 164 x^{10} + 2628 x^{9} - 2335 x^{8} - 3816 x^{7} + 5792 x^{6} + 460 x^{5} - 4152 x^{4} + 2060 x^{3} - 198 x^{2} - 36 x - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(191958112137556655079424=2^{32}\cdot 41^{6}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.52$
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:  $2, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{3}{10} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{3}{10} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{10} a^{9} + \frac{1}{10} a^{8} + \frac{3}{10} a^{7} + \frac{1}{5} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} + \frac{1}{10} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{10}$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{3}{10} a^{5} - \frac{1}{5} a^{4} + \frac{1}{10} a^{3} + \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{185830} a^{15} - \frac{1572}{92915} a^{14} + \frac{4569}{185830} a^{13} - \frac{248}{92915} a^{12} - \frac{1723}{92915} a^{11} + \frac{4158}{18583} a^{10} + \frac{4341}{37166} a^{9} - \frac{7211}{185830} a^{8} - \frac{24819}{92915} a^{7} + \frac{11233}{92915} a^{6} - \frac{14963}{185830} a^{5} + \frac{25039}{185830} a^{4} - \frac{7821}{37166} a^{3} - \frac{39518}{92915} a^{2} + \frac{1546}{18583} a + \frac{12943}{185830}$

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:  $15$
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:  \( 2112745.57805 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T876:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 65 conjugacy class representatives for t16n876 are not computed
Character table for t16n876 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.282300416.1, 8.8.10686103552.1, 8.8.438130245632.1

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

Sibling fields

Degree 16 siblings: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$