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Properties

Label 16.16.1022187390...8125.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{8}\cdot 13^{10}\cdot 29^{7}\cdot 1049^{2}$
Root discriminant $115.64$
Ramified primes $5, 13, 29, 1049$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1558

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-120710321, 1942920230, 1392532762, -2428250331, -173717859, 716474892, -73978895, -75326536, 12415935, 3680393, -734082, -88305, 20060, 946, -246, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 246*x^14 + 946*x^13 + 20060*x^12 - 88305*x^11 - 734082*x^10 + 3680393*x^9 + 12415935*x^8 - 75326536*x^7 - 73978895*x^6 + 716474892*x^5 - 173717859*x^4 - 2428250331*x^3 + 1392532762*x^2 + 1942920230*x - 120710321)
 
gp: K = bnfinit(x^16 - 3*x^15 - 246*x^14 + 946*x^13 + 20060*x^12 - 88305*x^11 - 734082*x^10 + 3680393*x^9 + 12415935*x^8 - 75326536*x^7 - 73978895*x^6 + 716474892*x^5 - 173717859*x^4 - 2428250331*x^3 + 1392532762*x^2 + 1942920230*x - 120710321, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 246 x^{14} + 946 x^{13} + 20060 x^{12} - 88305 x^{11} - 734082 x^{10} + 3680393 x^{9} + 12415935 x^{8} - 75326536 x^{7} - 73978895 x^{6} + 716474892 x^{5} - 173717859 x^{4} - 2428250331 x^{3} + 1392532762 x^{2} + 1942920230 x - 120710321 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1022187390863100363395289742578125=5^{8}\cdot 13^{10}\cdot 29^{7}\cdot 1049^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.64$
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:  $5, 13, 29, 1049$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{55} a^{14} - \frac{17}{55} a^{13} - \frac{7}{55} a^{12} - \frac{18}{55} a^{11} + \frac{2}{11} a^{10} - \frac{23}{55} a^{9} + \frac{1}{11} a^{8} - \frac{5}{11} a^{7} - \frac{3}{11} a^{6} - \frac{1}{55} a^{5} + \frac{9}{55} a^{4} - \frac{1}{11} a^{3} - \frac{1}{11} a^{2} + \frac{9}{55} a + \frac{6}{55}$, $\frac{1}{6681277402156549510937794737058439450184333362310714542327455} a^{15} + \frac{26251415764772073085293216612594618928501895964622758481869}{6681277402156549510937794737058439450184333362310714542327455} a^{14} + \frac{2949572205510940509994606844374775407091578830966801902489031}{6681277402156549510937794737058439450184333362310714542327455} a^{13} + \frac{535713924474747579621483918504351385178324143645479979266410}{1336255480431309902187558947411687890036866672462142908465491} a^{12} - \frac{3087958519462743077337098658227300000263725407160171352253353}{6681277402156549510937794737058439450184333362310714542327455} a^{11} + \frac{184546950103841750665442571322049203661371935798509256580202}{6681277402156549510937794737058439450184333362310714542327455} a^{10} + \frac{2260763595054943528354583209422708739168962486626985549070607}{6681277402156549510937794737058439450184333362310714542327455} a^{9} + \frac{23498171731672453447777615056162904818532528920478649993777}{70329235812174205378292576179562520528256140655902258340289} a^{8} - \frac{95787542613809859111648834508947404437721175176500255925592}{1336255480431309902187558947411687890036866672462142908465491} a^{7} - \frac{1951359248238042304336695201043327138309184226941604301247161}{6681277402156549510937794737058439450184333362310714542327455} a^{6} - \frac{2502652001094677798169555700825890999230215971361846364344812}{6681277402156549510937794737058439450184333362310714542327455} a^{5} + \frac{1382105792537292360154740314159182276489694297251656565501094}{6681277402156549510937794737058439450184333362310714542327455} a^{4} - \frac{71553028492785242232566537212091777838595658325082000633385}{1336255480431309902187558947411687890036866672462142908465491} a^{3} - \frac{160929501413216314719642770097440017187311985349197459270266}{607388854741504500994344976096221768198575760210064958393405} a^{2} + \frac{286766067312318506222380079210300962879642378551305770902467}{1336255480431309902187558947411687890036866672462142908465491} a + \frac{2056132534446983230203121009840412295810348957856745080129841}{6681277402156549510937794737058439450184333362310714542327455}$

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:  $15$
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:  \( 91596632835.7 \) (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, 8.8.2576088125.1

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 $16$ $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ R $16$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.8.6.4$x^{8} - 13 x^{4} + 338$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
13.8.4.2$x^{8} + 169 x^{4} - 2197 x^{2} + 57122$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$29$29.8.7.3$x^{8} + 58$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
1049Data not computed