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Properties

Label 16.0.87071293440...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{10}$
Root discriminant $17.63$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $(C_4\times C_8):C_2$ (as 16T114)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, -338, 1079, -2094, 2877, -3154, 2836, -2132, 1405, -792, 432, -194, 93, -30, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 13*x^14 - 30*x^13 + 93*x^12 - 194*x^11 + 432*x^10 - 792*x^9 + 1405*x^8 - 2132*x^7 + 2836*x^6 - 3154*x^5 + 2877*x^4 - 2094*x^3 + 1079*x^2 - 338*x + 49)
 
gp: K = bnfinit(x^16 - 2*x^15 + 13*x^14 - 30*x^13 + 93*x^12 - 194*x^11 + 432*x^10 - 792*x^9 + 1405*x^8 - 2132*x^7 + 2836*x^6 - 3154*x^5 + 2877*x^4 - 2094*x^3 + 1079*x^2 - 338*x + 49, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 13 x^{14} - 30 x^{13} + 93 x^{12} - 194 x^{11} + 432 x^{10} - 792 x^{9} + 1405 x^{8} - 2132 x^{7} + 2836 x^{6} - 3154 x^{5} + 2877 x^{4} - 2094 x^{3} + 1079 x^{2} - 338 x + 49 \)

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:  \(87071293440000000000=2^{24}\cdot 3^{12}\cdot 5^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.63$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{3}{7} a^{13} - \frac{3}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{87494198807029} a^{15} + \frac{3991829672123}{87494198807029} a^{14} - \frac{17311857521527}{87494198807029} a^{13} + \frac{6309317806620}{87494198807029} a^{12} - \frac{23238458211341}{87494198807029} a^{11} + \frac{5033192533890}{12499171258147} a^{10} + \frac{19694936830364}{87494198807029} a^{9} + \frac{27191889042840}{87494198807029} a^{8} - \frac{14196289389504}{87494198807029} a^{7} - \frac{16614799903354}{87494198807029} a^{6} + \frac{5046906938380}{87494198807029} a^{5} - \frac{5641612755917}{87494198807029} a^{4} + \frac{40155416098632}{87494198807029} a^{3} + \frac{20494775119729}{87494198807029} a^{2} + \frac{5691226661717}{87494198807029} a - \frac{384685605023}{1785595894021}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{3784467282}{431339503} a^{15} - \frac{5033084806}{431339503} a^{14} + \frac{45822138150}{431339503} a^{13} - \frac{82830499518}{431339503} a^{12} + \frac{296415049040}{431339503} a^{11} - \frac{76506642239}{61619929} a^{10} + \frac{1275838724620}{431339503} a^{9} - \frac{2142203113543}{431339503} a^{8} + \frac{3881038358038}{431339503} a^{7} - \frac{5467080921024}{431339503} a^{6} + \frac{7067558043846}{431339503} a^{5} - \frac{7198434673491}{431339503} a^{4} + \frac{6061747508728}{431339503} a^{3} - \frac{3860456614958}{431339503} a^{2} + \frac{1494645655596}{431339503} a - \frac{5635864523}{8802847} \) (order $6$)
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:  \( 5789.2337068 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_4\times C_8):C_2$ (as 16T114):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_4\times C_8):C_2$
Character table for $(C_4\times C_8):C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 8.0.8294400.1

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

Sibling fields

Degree 16 sibling: 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
3Data not computed
$5$5.8.4.2$x^{8} + 25 x^{4} - 250 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5.8.6.4$x^{8} - 5 x^{4} + 50$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$