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Properties

Label 16.0.73286708086...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 547889^{2}$
Root discriminant $17.44$
Ramified primes $5, 547889$
Class number $1$
Class group Trivial
Galois group 16T1497

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -896, 1536, -1728, 1344, -584, -136, 486, -457, 243, -34, -73, 84, -54, 24, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 24*x^14 - 54*x^13 + 84*x^12 - 73*x^11 - 34*x^10 + 243*x^9 - 457*x^8 + 486*x^7 - 136*x^6 - 584*x^5 + 1344*x^4 - 1728*x^3 + 1536*x^2 - 896*x + 256)
 
gp: K = bnfinit(x^16 - 7*x^15 + 24*x^14 - 54*x^13 + 84*x^12 - 73*x^11 - 34*x^10 + 243*x^9 - 457*x^8 + 486*x^7 - 136*x^6 - 584*x^5 + 1344*x^4 - 1728*x^3 + 1536*x^2 - 896*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 24 x^{14} - 54 x^{13} + 84 x^{12} - 73 x^{11} - 34 x^{10} + 243 x^{9} - 457 x^{8} + 486 x^{7} - 136 x^{6} - 584 x^{5} + 1344 x^{4} - 1728 x^{3} + 1536 x^{2} - 896 x + 256 \)

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:  \(73286708086181640625=5^{12}\cdot 547889^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.44$
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:  $5, 547889$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{2} a^{8} - \frac{1}{16} a^{7} - \frac{1}{2} a^{6} + \frac{7}{16} a^{5} + \frac{1}{16} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{4} a^{9} - \frac{1}{32} a^{8} - \frac{1}{4} a^{7} - \frac{9}{32} a^{6} - \frac{15}{32} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{13} - \frac{1}{32} a^{12} + \frac{1}{32} a^{11} - \frac{1}{8} a^{10} - \frac{1}{64} a^{9} - \frac{1}{8} a^{8} + \frac{23}{64} a^{7} - \frac{15}{64} a^{6} - \frac{1}{2} a^{5} + \frac{5}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{128} a^{15} - \frac{1}{128} a^{14} - \frac{1}{64} a^{13} + \frac{1}{64} a^{12} - \frac{1}{16} a^{11} - \frac{1}{128} a^{10} - \frac{1}{16} a^{9} - \frac{41}{128} a^{8} - \frac{15}{128} a^{7} + \frac{1}{4} a^{6} + \frac{5}{32} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$

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:  \( -\frac{23}{128} a^{15} + \frac{141}{128} a^{14} - \frac{105}{32} a^{13} + \frac{413}{64} a^{12} - \frac{275}{32} a^{11} + \frac{559}{128} a^{10} + \frac{697}{64} a^{9} - \frac{4229}{128} a^{8} + \frac{6235}{128} a^{7} - \frac{2447}{64} a^{6} - \frac{57}{4} a^{5} + \frac{367}{4} a^{4} - \frac{1199}{8} a^{3} + \frac{321}{2} a^{2} - 117 a + 43 \) (order $10$)
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:  \( 9516.84702281 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1497:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1497
Character table for t16n1497 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.8.1712153125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
547889Data not computed