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Properties

Label 16.2.12982767816...6875.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,5^{13}\cdot 41^{6}\cdot 2239$
Root discriminant $24.10$
Ramified primes $5, 41, 2239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1774

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, 63, -280, 480, -10, -896, 1187, -825, 345, 30, -133, 36, 20, -10, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 5*x^14 - 10*x^13 + 20*x^12 + 36*x^11 - 133*x^10 + 30*x^9 + 345*x^8 - 825*x^7 + 1187*x^6 - 896*x^5 - 10*x^4 + 480*x^3 - 280*x^2 + 63*x - 9)
 
gp: K = bnfinit(x^16 - 3*x^15 + 5*x^14 - 10*x^13 + 20*x^12 + 36*x^11 - 133*x^10 + 30*x^9 + 345*x^8 - 825*x^7 + 1187*x^6 - 896*x^5 - 10*x^4 + 480*x^3 - 280*x^2 + 63*x - 9, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 5 x^{14} - 10 x^{13} + 20 x^{12} + 36 x^{11} - 133 x^{10} + 30 x^{9} + 345 x^{8} - 825 x^{7} + 1187 x^{6} - 896 x^{5} - 10 x^{4} + 480 x^{3} - 280 x^{2} + 63 x - 9 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-12982767816893310546875=-\,5^{13}\cdot 41^{6}\cdot 2239\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.10$
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:  $5, 41, 2239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{25} a^{12} - \frac{1}{25} a^{11} - \frac{1}{25} a^{10} - \frac{6}{25} a^{7} - \frac{4}{25} a^{6} - \frac{4}{25} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{4}{25} a^{2} + \frac{6}{25} a + \frac{1}{25}$, $\frac{1}{25} a^{13} - \frac{2}{25} a^{11} - \frac{1}{25} a^{10} - \frac{1}{25} a^{8} - \frac{1}{5} a^{7} + \frac{2}{25} a^{6} - \frac{9}{25} a^{5} - \frac{1}{5} a^{4} + \frac{4}{25} a^{3} - \frac{1}{5} a^{2} + \frac{2}{25} a + \frac{6}{25}$, $\frac{1}{25} a^{14} + \frac{2}{25} a^{11} - \frac{2}{25} a^{10} - \frac{1}{25} a^{9} - \frac{1}{5} a^{7} - \frac{12}{25} a^{6} + \frac{2}{25} a^{5} - \frac{11}{25} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{8}{25} a + \frac{7}{25}$, $\frac{1}{565309240125} a^{15} + \frac{66145897}{37687282675} a^{14} + \frac{2131336867}{113061848025} a^{13} + \frac{55355768}{22612369605} a^{12} + \frac{2096082997}{113061848025} a^{11} + \frac{6638851932}{188436413375} a^{10} - \frac{7137395108}{113061848025} a^{9} + \frac{604925493}{37687282675} a^{8} + \frac{371767237}{7537456535} a^{7} + \frac{1138524537}{5383897525} a^{6} + \frac{242535756887}{565309240125} a^{5} - \frac{5904625046}{22612369605} a^{4} - \frac{518909684}{4522473921} a^{3} - \frac{2054066974}{7537456535} a^{2} - \frac{55004333519}{113061848025} a - \frac{16640293259}{188436413375}$

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:  \( 81840.2756508 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1774:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 130 conjugacy class representatives for t16n1774 are not computed
Character table for t16n1774 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5125.1, 8.4.1076890625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
2239Data not computed