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Properties

Label 16.0.79215868954...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 19^{8}\cdot 103^{6}$
Root discriminant $55.42$
Ramified primes $5, 19, 103$
Class number $1616$ (GRH)
Class group $[2, 2, 404]$ (GRH)
Galois group $Q_8:C_2^2.D_6$ (as 16T754)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![226071, -553149, 842431, -830694, 741836, -532544, 376822, -215474, 122896, -54193, 23397, -6903, 2138, -400, 82, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 82*x^14 - 400*x^13 + 2138*x^12 - 6903*x^11 + 23397*x^10 - 54193*x^9 + 122896*x^8 - 215474*x^7 + 376822*x^6 - 532544*x^5 + 741836*x^4 - 830694*x^3 + 842431*x^2 - 553149*x + 226071)
 
gp: K = bnfinit(x^16 - 8*x^15 + 82*x^14 - 400*x^13 + 2138*x^12 - 6903*x^11 + 23397*x^10 - 54193*x^9 + 122896*x^8 - 215474*x^7 + 376822*x^6 - 532544*x^5 + 741836*x^4 - 830694*x^3 + 842431*x^2 - 553149*x + 226071, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 82 x^{14} - 400 x^{13} + 2138 x^{12} - 6903 x^{11} + 23397 x^{10} - 54193 x^{9} + 122896 x^{8} - 215474 x^{7} + 376822 x^{6} - 532544 x^{5} + 741836 x^{4} - 830694 x^{3} + 842431 x^{2} - 553149 x + 226071 \)

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:  \(7921586895449647259644140625=5^{8}\cdot 19^{8}\cdot 103^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.42$
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:  $5, 19, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{6066623318377244163057414835382340164598291} a^{15} - \frac{292341105295227545684399280041516716568750}{6066623318377244163057414835382340164598291} a^{14} - \frac{63449286375711662392633482688196022391735}{356860195198661421356318519728372950858723} a^{13} - \frac{601006619123254544053887880939365090002176}{6066623318377244163057414835382340164598291} a^{12} - \frac{1828809181806651806481777867303917056193743}{6066623318377244163057414835382340164598291} a^{11} - \frac{295167730497895091028000884940191494343665}{674069257597471573673046092820260018288699} a^{10} + \frac{168812020448075466932253545428154168358476}{2022207772792414721019138278460780054866097} a^{9} + \frac{1396084811391049734620350118072239270113059}{6066623318377244163057414835382340164598291} a^{8} - \frac{385827795590085577727369954124381012732029}{6066623318377244163057414835382340164598291} a^{7} + \frac{248574147431698502188313301862348850887117}{6066623318377244163057414835382340164598291} a^{6} - \frac{511136545678247691128014685583988827887657}{6066623318377244163057414835382340164598291} a^{5} + \frac{823549826656275625343852140328200491460132}{6066623318377244163057414835382340164598291} a^{4} - \frac{31225277349156139701383359332860136479215}{6066623318377244163057414835382340164598291} a^{3} + \frac{558571434792220954302474396244555131787109}{2022207772792414721019138278460780054866097} a^{2} - \frac{2744673576228411054503255750614707410987120}{6066623318377244163057414835382340164598291} a + \frac{171506337558300115095442200607851896516584}{674069257597471573673046092820260018288699}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{404}$, which has order $1616$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10491.0986114 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2.D_6$ (as 16T754):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $Q_8:C_2^2.D_6$
Character table for $Q_8:C_2^2.D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.1957.1, 8.8.2393655625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$103$103.4.2.2$x^{4} - 103 x^{2} + 53045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.8.4.1$x^{8} + 106090 x^{4} - 1092727 x^{2} + 2813772025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$