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Properties

Label 16.0.83936997636...3125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{6}\cdot 1279\cdot 5659$
Root discriminant $36.12$
Ramified primes $5, 41, 1279, 5659$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1774

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16871, -17560, 22252, -26098, 13207, -6012, 3324, 2221, -1260, 239, -258, 1, 57, -22, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 12*x^14 - 22*x^13 + 57*x^12 + x^11 - 258*x^10 + 239*x^9 - 1260*x^8 + 2221*x^7 + 3324*x^6 - 6012*x^5 + 13207*x^4 - 26098*x^3 + 22252*x^2 - 17560*x + 16871)
 
gp: K = bnfinit(x^16 - 4*x^15 + 12*x^14 - 22*x^13 + 57*x^12 + x^11 - 258*x^10 + 239*x^9 - 1260*x^8 + 2221*x^7 + 3324*x^6 - 6012*x^5 + 13207*x^4 - 26098*x^3 + 22252*x^2 - 17560*x + 16871, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 12 x^{14} - 22 x^{13} + 57 x^{12} + x^{11} - 258 x^{10} + 239 x^{9} - 1260 x^{8} + 2221 x^{7} + 3324 x^{6} - 6012 x^{5} + 13207 x^{4} - 26098 x^{3} + 22252 x^{2} - 17560 x + 16871 \)

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:  \(8393699763639770751953125=5^{12}\cdot 41^{6}\cdot 1279\cdot 5659\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.12$
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:  $5, 41, 1279, 5659$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{38} a^{14} - \frac{1}{38} a^{13} - \frac{3}{19} a^{12} + \frac{13}{38} a^{11} - \frac{2}{19} a^{10} - \frac{8}{19} a^{9} - \frac{9}{19} a^{8} + \frac{7}{38} a^{7} - \frac{1}{2} a^{6} + \frac{7}{38} a^{5} - \frac{9}{19} a^{4} - \frac{15}{38} a^{3} + \frac{9}{19} a^{2} - \frac{17}{38} a + \frac{5}{38}$, $\frac{1}{90177514690248760467041210063902} a^{15} - \frac{73169603800561428509329766949}{90177514690248760467041210063902} a^{14} + \frac{17587996497674637428560455164915}{45088757345124380233520605031951} a^{13} - \frac{11400265136617842161090548174095}{90177514690248760467041210063902} a^{12} - \frac{4428422745533605821126770161049}{45088757345124380233520605031951} a^{11} - \frac{11281325172740717255746587404}{48430459017319420229345440421} a^{10} + \frac{12443562573470395974173678351909}{45088757345124380233520605031951} a^{9} - \frac{19313207737983987370748402332671}{90177514690248760467041210063902} a^{8} + \frac{23209924336824180454842928632501}{90177514690248760467041210063902} a^{7} - \frac{2543110923020877664761072032501}{12882502098606965781005887151986} a^{6} - \frac{17475926366904001892280190169039}{45088757345124380233520605031951} a^{5} + \frac{2215737287722122538830542820241}{4746184983697303182475853161258} a^{4} + \frac{17940299777427228270784306771639}{45088757345124380233520605031951} a^{3} + \frac{6335570846611749549928535656407}{90177514690248760467041210063902} a^{2} + \frac{5064980819018674801918710656347}{12882502098606965781005887151986} a - \frac{1151019684122976800710483036798}{45088757345124380233520605031951}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 337558.454715 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
1279Data not computed
5659Data not computed