# Properties

 Label 16.0.5190614520439741.1 Degree $16$ Signature $[0, 8]$ Discriminant $617^{2}\cdot 2389^{3}$ Root discriminant $9.60$ Ramified primes $617, 2389$ Class number $1$ Class group Trivial Galois Group 16T1948

# Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -1, -5, 3, 6, -5, 4, -2, -5, 12, -15, 15, -12, 7, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 15*x^12 - 15*x^11 + 12*x^10 - 5*x^9 - 2*x^8 + 4*x^7 - 5*x^6 + 6*x^5 + 3*x^4 - 5*x^3 - x^2 + 1)
gp: K = bnfinit(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 15*x^12 - 15*x^11 + 12*x^10 - 5*x^9 - 2*x^8 + 4*x^7 - 5*x^6 + 6*x^5 + 3*x^4 - 5*x^3 - x^2 + 1, 1)

## Normalizeddefining polynomial

$$x^{16}$$ $$\mathstrut -\mathstrut 3 x^{15}$$ $$\mathstrut +\mathstrut 7 x^{14}$$ $$\mathstrut -\mathstrut 12 x^{13}$$ $$\mathstrut +\mathstrut 15 x^{12}$$ $$\mathstrut -\mathstrut 15 x^{11}$$ $$\mathstrut +\mathstrut 12 x^{10}$$ $$\mathstrut -\mathstrut 5 x^{9}$$ $$\mathstrut -\mathstrut 2 x^{8}$$ $$\mathstrut +\mathstrut 4 x^{7}$$ $$\mathstrut -\mathstrut 5 x^{6}$$ $$\mathstrut +\mathstrut 6 x^{5}$$ $$\mathstrut +\mathstrut 3 x^{4}$$ $$\mathstrut -\mathstrut 5 x^{3}$$ $$\mathstrut -\mathstrut x^{2}$$ $$\mathstrut +\mathstrut 1$$

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

## Invariants

 Degree: $16$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[0, 8]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$5190614520439741=617^{2}\cdot 2389^{3}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $9.60$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $617, 2389$ magma: PrimeDivisors(Discriminant(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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{692467} a^{15} - \frac{330365}{692467} a^{14} + \frac{318267}{692467} a^{13} + \frac{174147}{692467} a^{12} - \frac{7905}{692467} a^{11} + \frac{218538}{692467} a^{10} - \frac{41324}{692467} a^{9} - \frac{107622}{692467} a^{8} + \frac{193514}{692467} a^{7} + \frac{266310}{692467} a^{6} - \frac{79408}{692467} a^{5} - \frac{34126}{692467} a^{4} - \frac{121612}{692467} a^{3} - \frac{259334}{692467} a^{2} + \frac{4266}{692467} a - \frac{153947}{692467}$

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: $$-1$$ (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 magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$10.0160978095$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A non-solvable group of order 10321920 The 185 conjugacy class representatives for t16n1948 are not computed Character table for t16n1948 is not computed

## Intermediate fields

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ $16$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
617Data not computed
2389Data not computed