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Properties

Label 16.0.95021156485...2509.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 79^{3}\cdot 331^{3}$
Root discriminant $15.35$
Ramified primes $3, 79, 331$
Class number $1$
Class group Trivial
Galois group 16T1871

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, -21, 113, -69, 86, -144, 128, -216, 250, -240, 220, -156, 95, -48, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 19*x^14 - 48*x^13 + 95*x^12 - 156*x^11 + 220*x^10 - 240*x^9 + 250*x^8 - 216*x^7 + 128*x^6 - 144*x^5 + 86*x^4 - 69*x^3 + 113*x^2 - 21*x + 49)
 
gp: K = bnfinit(x^16 - 6*x^15 + 19*x^14 - 48*x^13 + 95*x^12 - 156*x^11 + 220*x^10 - 240*x^9 + 250*x^8 - 216*x^7 + 128*x^6 - 144*x^5 + 86*x^4 - 69*x^3 + 113*x^2 - 21*x + 49, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 19 x^{14} - 48 x^{13} + 95 x^{12} - 156 x^{11} + 220 x^{10} - 240 x^{9} + 250 x^{8} - 216 x^{7} + 128 x^{6} - 144 x^{5} + 86 x^{4} - 69 x^{3} + 113 x^{2} - 21 x + 49 \)

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:  \(9502115648519722509=3^{12}\cdot 79^{3}\cdot 331^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.35$
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:  $3, 79, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{88320425291} a^{15} + \frac{3715000111}{88320425291} a^{14} + \frac{461264436}{88320425291} a^{13} - \frac{12598547329}{88320425291} a^{12} - \frac{19080743949}{88320425291} a^{11} + \frac{3272503117}{88320425291} a^{10} + \frac{10122227096}{88320425291} a^{9} + \frac{41419764082}{88320425291} a^{8} - \frac{17232396423}{88320425291} a^{7} - \frac{9480070667}{88320425291} a^{6} + \frac{20083733219}{88320425291} a^{5} - \frac{26120230513}{88320425291} a^{4} - \frac{33485102317}{88320425291} a^{3} + \frac{20023408533}{88320425291} a^{2} + \frac{236745701}{1802457659} a - \frac{599132240}{1802457659}$

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:  \( -\frac{799418}{16517753} a^{15} + \frac{3454218}{16517753} a^{14} - \frac{5670894}{16517753} a^{13} + \frac{4900863}{16517753} a^{12} + \frac{8494133}{16517753} a^{11} - \frac{41160904}{16517753} a^{10} + \frac{87008543}{16517753} a^{9} - \frac{152003980}{16517753} a^{8} + \frac{144760933}{16517753} a^{7} - \frac{103035672}{16517753} a^{6} + \frac{101776127}{16517753} a^{5} - \frac{6627905}{16517753} a^{4} + \frac{20375671}{16517753} a^{3} - \frac{72386742}{16517753} a^{2} + \frac{3778449}{2359679} a - \frac{978229}{337097} \) (order $6$)
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:  \( 1994.62365846 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1871:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 104 conjugacy class representatives for t16n1871 are not computed
Character table for t16n1871 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 8.0.2118069.1

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 $16$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $16$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
79Data not computed
331Data not computed