↑

Properties

Label 16.8.14085713869...0625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 13^{8}\cdot 29^{4}$
Root discriminant $27.98$
Ramified primes $5, 13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3$ (as 16T217)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, -66, -113, 16, 453, 366, 148, -374, -571, -30, -50, 232, -18, -10, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 10*x^13 - 18*x^12 + 232*x^11 - 50*x^10 - 30*x^9 - 571*x^8 - 374*x^7 + 148*x^6 + 366*x^5 + 453*x^4 + 16*x^3 - 113*x^2 - 66*x - 9)
 
gp: K = bnfinit(x^16 - 4*x^15 - 10*x^13 - 18*x^12 + 232*x^11 - 50*x^10 - 30*x^9 - 571*x^8 - 374*x^7 + 148*x^6 + 366*x^5 + 453*x^4 + 16*x^3 - 113*x^2 - 66*x - 9, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 10 x^{13} - 18 x^{12} + 232 x^{11} - 50 x^{10} - 30 x^{9} - 571 x^{8} - 374 x^{7} + 148 x^{6} + 366 x^{5} + 453 x^{4} + 16 x^{3} - 113 x^{2} - 66 x - 9 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(140857138691308837890625=5^{12}\cdot 13^{8}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.98$
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:  $5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{10802718326831511918} a^{15} - \frac{21569607402714309}{1800453054471918653} a^{14} + \frac{296050371843961283}{10802718326831511918} a^{13} - \frac{422577027504656431}{1800453054471918653} a^{12} - \frac{364448587724236352}{5401359163415755959} a^{11} + \frac{196016535424331627}{10802718326831511918} a^{10} + \frac{555604531513942093}{5401359163415755959} a^{9} + \frac{100667799461935303}{635454019225383054} a^{8} - \frac{76711178902774901}{10802718326831511918} a^{7} + \frac{2508777304642918709}{5401359163415755959} a^{6} - \frac{1785643038094005386}{5401359163415755959} a^{5} - \frac{2299123121624293594}{5401359163415755959} a^{4} - \frac{2346371967637827760}{5401359163415755959} a^{3} + \frac{1281427023267057437}{3600906108943837306} a^{2} + \frac{2134916876531101510}{5401359163415755959} a - \frac{766484507938611548}{1800453054471918653}$

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:  $11$
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:  \( 570935.403749 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T217):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}, \sqrt{13})\), 8.4.517668125.1, 8.4.12941703125.1, 8.8.375309390625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$