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Properties

Label 16.0.14388016060...3125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{4}\cdot 941^{3}$
Root discriminant $28.01$
Ramified primes $5, 29, 941$
Class number $4$
Class group $[4]$
Galois group 16T1561

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, 798, 49, -31, 1515, 436, -111, -45, 27, -274, 91, -28, 2, -5, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 6*x^14 - 5*x^13 + 2*x^12 - 28*x^11 + 91*x^10 - 274*x^9 + 27*x^8 - 45*x^7 - 111*x^6 + 436*x^5 + 1515*x^4 - 31*x^3 + 49*x^2 + 798*x + 361)
 
gp: K = bnfinit(x^16 - x^15 + 6*x^14 - 5*x^13 + 2*x^12 - 28*x^11 + 91*x^10 - 274*x^9 + 27*x^8 - 45*x^7 - 111*x^6 + 436*x^5 + 1515*x^4 - 31*x^3 + 49*x^2 + 798*x + 361, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 6 x^{14} - 5 x^{13} + 2 x^{12} - 28 x^{11} + 91 x^{10} - 274 x^{9} + 27 x^{8} - 45 x^{7} - 111 x^{6} + 436 x^{5} + 1515 x^{4} - 31 x^{3} + 49 x^{2} + 798 x + 361 \)

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:  \(143880160600219970703125=5^{12}\cdot 29^{4}\cdot 941^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.01$
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:  $5, 29, 941$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} + \frac{4}{19} a^{12} + \frac{5}{19} a^{10} + \frac{8}{19} a^{9} - \frac{4}{19} a^{8} - \frac{4}{19} a^{7} - \frac{2}{19} a^{5} + \frac{2}{19} a^{4} + \frac{8}{19} a^{3} + \frac{6}{19} a^{2} + \frac{7}{19} a$, $\frac{1}{19} a^{14} + \frac{3}{19} a^{12} + \frac{5}{19} a^{11} + \frac{7}{19} a^{10} + \frac{2}{19} a^{9} - \frac{7}{19} a^{8} - \frac{3}{19} a^{7} - \frac{2}{19} a^{6} - \frac{9}{19} a^{5} - \frac{7}{19} a^{3} + \frac{2}{19} a^{2} - \frac{9}{19} a$, $\frac{1}{1079134948615920160013329} a^{15} + \frac{15853036668615995525106}{1079134948615920160013329} a^{14} + \frac{16476422625243392238935}{1079134948615920160013329} a^{13} + \frac{211275515694075817802577}{1079134948615920160013329} a^{12} - \frac{407552016313552384130489}{1079134948615920160013329} a^{11} + \frac{52979032548601694562869}{1079134948615920160013329} a^{10} - \frac{483638439476365727418472}{1079134948615920160013329} a^{9} + \frac{9608553109840868713105}{56796576242943166316491} a^{8} + \frac{243187170327789426567406}{1079134948615920160013329} a^{7} - \frac{86468307439950793182274}{1079134948615920160013329} a^{6} - \frac{452773030677561620391151}{1079134948615920160013329} a^{5} + \frac{81316395802138822244980}{1079134948615920160013329} a^{4} + \frac{294549454303604565273451}{1079134948615920160013329} a^{3} - \frac{427209427897821752094461}{1079134948615920160013329} a^{2} + \frac{46395731277517123420574}{1079134948615920160013329} a - \frac{22498253818527656766523}{56796576242943166316491}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15225.1671836 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1561:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1561 are not computed
Character table for t16n1561 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.0.12365328125.2

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 $16$ $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
$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}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$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}$
941Data not computed