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Properties

Label 16.4.58264304400...0000.4
Degree $16$
Signature $[4, 6]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{14}\cdot 149^{2}$
Root discriminant $26.48$
Ramified primes $2, 3, 5, 149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1720

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, -176, 293, 228, -1584, 2674, -2445, 1244, 209, -1102, 1195, -828, 411, -154, 42, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 42*x^14 - 154*x^13 + 411*x^12 - 828*x^11 + 1195*x^10 - 1102*x^9 + 209*x^8 + 1244*x^7 - 2445*x^6 + 2674*x^5 - 1584*x^4 + 228*x^3 + 293*x^2 - 176*x + 31)
 
gp: K = bnfinit(x^16 - 8*x^15 + 42*x^14 - 154*x^13 + 411*x^12 - 828*x^11 + 1195*x^10 - 1102*x^9 + 209*x^8 + 1244*x^7 - 2445*x^6 + 2674*x^5 - 1584*x^4 + 228*x^3 + 293*x^2 - 176*x + 31, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 42 x^{14} - 154 x^{13} + 411 x^{12} - 828 x^{11} + 1195 x^{10} - 1102 x^{9} + 209 x^{8} + 1244 x^{7} - 2445 x^{6} + 2674 x^{5} - 1584 x^{4} + 228 x^{3} + 293 x^{2} - 176 x + 31 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(58264304400000000000000=2^{16}\cdot 3^{8}\cdot 5^{14}\cdot 149^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.48$
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:  $2, 3, 5, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{5} - \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{6} - \frac{2}{5} a$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{7} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{18922255} a^{14} - \frac{7}{18922255} a^{13} - \frac{370340}{3784451} a^{12} - \frac{243062}{18922255} a^{11} + \frac{94701}{1720205} a^{10} - \frac{98948}{1720205} a^{9} - \frac{1286608}{18922255} a^{8} - \frac{2349749}{18922255} a^{7} + \frac{2786153}{18922255} a^{6} - \frac{47143}{1720205} a^{5} + \frac{679879}{3784451} a^{4} + \frac{6617}{344041} a^{3} + \frac{1751881}{3784451} a^{2} + \frac{6125331}{18922255} a - \frac{1122124}{3784451}$, $\frac{1}{18922255} a^{15} - \frac{1851749}{18922255} a^{13} - \frac{1851609}{18922255} a^{12} - \frac{659723}{18922255} a^{11} - \frac{124123}{1720205} a^{10} - \frac{1336702}{18922255} a^{9} - \frac{2652}{18922255} a^{8} + \frac{1475714}{18922255} a^{7} - \frac{7506659}{18922255} a^{6} - \frac{4015067}{18922255} a^{5} + \frac{9021896}{18922255} a^{4} + \frac{7522499}{18922255} a^{3} - \frac{8247854}{18922255} a^{2} - \frac{4362264}{18922255} a - \frac{285966}{3784451}$

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:  $9$
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:  \( 158578.946004 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1720:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 116 conjugacy class representatives for t16n1720 are not computed
Character table for t16n1720 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.1620000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
149Data not computed