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Properties

Label 16.0.54566491401...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}$
Root discriminant $72.20$
Ramified primes $2, 5, 13, 29$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1558

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![250591, -123442, -222246, 413450, -42224, -383612, 453076, -286190, 124109, -41668, 14360, -4418, 1266, -230, 42, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 42*x^14 - 230*x^13 + 1266*x^12 - 4418*x^11 + 14360*x^10 - 41668*x^9 + 124109*x^8 - 286190*x^7 + 453076*x^6 - 383612*x^5 - 42224*x^4 + 413450*x^3 - 222246*x^2 - 123442*x + 250591)
 
gp: K = bnfinit(x^16 - 4*x^15 + 42*x^14 - 230*x^13 + 1266*x^12 - 4418*x^11 + 14360*x^10 - 41668*x^9 + 124109*x^8 - 286190*x^7 + 453076*x^6 - 383612*x^5 - 42224*x^4 + 413450*x^3 - 222246*x^2 - 123442*x + 250591, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 42 x^{14} - 230 x^{13} + 1266 x^{12} - 4418 x^{11} + 14360 x^{10} - 41668 x^{9} + 124109 x^{8} - 286190 x^{7} + 453076 x^{6} - 383612 x^{5} - 42224 x^{4} + 413450 x^{3} - 222246 x^{2} - 123442 x + 250591 \)

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:  \(545664914012032080281600000000=2^{24}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.20$
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:  $2, 5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{22} a^{13} - \frac{5}{22} a^{12} + \frac{1}{11} a^{11} + \frac{2}{11} a^{10} - \frac{5}{22} a^{9} + \frac{5}{22} a^{8} - \frac{3}{11} a^{7} - \frac{5}{11} a^{6} + \frac{1}{22} a^{5} - \frac{5}{22} a^{4} - \frac{5}{11} a^{3} + \frac{3}{11} a^{2} - \frac{1}{11} a$, $\frac{1}{22} a^{14} - \frac{1}{22} a^{12} + \frac{3}{22} a^{11} + \frac{2}{11} a^{10} + \frac{1}{11} a^{9} - \frac{3}{22} a^{8} - \frac{7}{22} a^{7} + \frac{3}{11} a^{6} + \frac{9}{22} a^{4} - \frac{1}{2} a^{3} - \frac{5}{22} a^{2} - \frac{5}{11} a$, $\frac{1}{1603026568024763751338757040253995523137658} a^{15} + \frac{524342893465368447030966669680548016895}{801513284012381875669378520126997761568829} a^{14} + \frac{16223393976354822777549714186877188306714}{801513284012381875669378520126997761568829} a^{13} + \frac{156419260677962213510578216949420805157119}{1603026568024763751338757040253995523137658} a^{12} + \frac{216966489625211121280677726711438110533589}{1603026568024763751338757040253995523137658} a^{11} - \frac{97069659475460671873149642588137291541052}{801513284012381875669378520126997761568829} a^{10} - \frac{545388867622115677361989171603559193156}{72864844001125625060852592738817978324439} a^{9} - \frac{362044695422701729057250082856104699602359}{1603026568024763751338757040253995523137658} a^{8} + \frac{422415443508061032463989812975916573720413}{1603026568024763751338757040253995523137658} a^{7} - \frac{103517992798987765101738346205681270805017}{801513284012381875669378520126997761568829} a^{6} + \frac{306802622996277832219732866579281115337189}{801513284012381875669378520126997761568829} a^{5} - \frac{43138137754231494576857912269418277903499}{145729688002251250121705185477635956648878} a^{4} + \frac{174024788092379679681550799679485038752576}{801513284012381875669378520126997761568829} a^{3} + \frac{320000846036774821212698379186686016035569}{801513284012381875669378520126997761568829} a^{2} + \frac{102359700659558764311635560783995964507126}{801513284012381875669378520126997761568829} a - \frac{17246572096944675746049461843516705928450}{72864844001125625060852592738817978324439}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 17925777.4975 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1558:

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 t16n1558 are not computed
Character table for t16n1558 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.659478560000.2

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 R $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ R $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
13.8.6.4$x^{8} - 13 x^{4} + 338$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$29$29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
29.8.7.3$x^{8} + 58$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$