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Properties

Label 16.2.11094098739...0000.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,2^{8}\cdot 5^{10}\cdot 29^{4}\cdot 89^{4}$
Root discriminant $27.56$
Ramified primes $2, 5, 29, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1719

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, -33, -184, 392, -251, 265, -213, -7, 44, -87, 77, -48, 40, -15, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 9*x^14 - 15*x^13 + 40*x^12 - 48*x^11 + 77*x^10 - 87*x^9 + 44*x^8 - 7*x^7 - 213*x^6 + 265*x^5 - 251*x^4 + 392*x^3 - 184*x^2 - 33*x + 121)
 
gp: K = bnfinit(x^16 - 2*x^15 + 9*x^14 - 15*x^13 + 40*x^12 - 48*x^11 + 77*x^10 - 87*x^9 + 44*x^8 - 7*x^7 - 213*x^6 + 265*x^5 - 251*x^4 + 392*x^3 - 184*x^2 - 33*x + 121, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 9 x^{14} - 15 x^{13} + 40 x^{12} - 48 x^{11} + 77 x^{10} - 87 x^{9} + 44 x^{8} - 7 x^{7} - 213 x^{6} + 265 x^{5} - 251 x^{4} + 392 x^{3} - 184 x^{2} - 33 x + 121 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-110940987391802500000000=-\,2^{8}\cdot 5^{10}\cdot 29^{4}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.56$
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:  $2, 5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} + \frac{2}{11} a^{12} - \frac{3}{11} a^{11} - \frac{1}{11} a^{10} - \frac{3}{11} a^{9} + \frac{4}{11} a^{8} - \frac{1}{11} a^{7} + \frac{5}{11} a^{6} - \frac{4}{11} a^{5} - \frac{2}{11} a^{4} + \frac{2}{11} a^{3} + \frac{5}{11} a^{2} + \frac{4}{11} a$, $\frac{1}{11} a^{14} + \frac{4}{11} a^{12} + \frac{5}{11} a^{11} - \frac{1}{11} a^{10} - \frac{1}{11} a^{9} + \frac{2}{11} a^{8} - \frac{4}{11} a^{7} - \frac{3}{11} a^{6} - \frac{5}{11} a^{5} - \frac{5}{11} a^{4} + \frac{1}{11} a^{3} + \frac{5}{11} a^{2} + \frac{3}{11} a$, $\frac{1}{219780705281390143} a^{15} - \frac{8034135444691854}{219780705281390143} a^{14} - \frac{3424883090618754}{219780705281390143} a^{13} + \frac{109118208380215194}{219780705281390143} a^{12} - \frac{68285504144120161}{219780705281390143} a^{11} - \frac{58096649550713146}{219780705281390143} a^{10} - \frac{106992985459895885}{219780705281390143} a^{9} - \frac{57915948633593507}{219780705281390143} a^{8} - \frac{16747552853862394}{219780705281390143} a^{7} - \frac{10469334418007489}{219780705281390143} a^{6} + \frac{72158424677877720}{219780705281390143} a^{5} + \frac{88293693273679468}{219780705281390143} a^{4} - \frac{36237498642707358}{219780705281390143} a^{3} + \frac{20408799068204881}{219780705281390143} a^{2} + \frac{76304182221079808}{219780705281390143} a + \frac{9189602217741426}{19980064116490013}$

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:  $8$
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:  \( 98277.6806966 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1719:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 152 conjugacy class representatives for t16n1719 are not computed
Character table for t16n1719 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 4.4.64525.1, 4.4.725.1, 8.8.4163475625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$