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Properties

Label 16.16.1186139225...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{32}\cdot 5^{14}\cdot 199^{2}\cdot 421^{2}\cdot 2539^{2}$
Root discriminant $179.74$
Ramified primes $2, 5, 199, 421, 2539$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1162

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![624192924176, 254794977984, -215062276912, -76367046112, 30570952056, 8765075664, -2314815140, -500712216, 101165629, 15509448, -2607840, -263408, 38866, 2296, -308, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 308*x^14 + 2296*x^13 + 38866*x^12 - 263408*x^11 - 2607840*x^10 + 15509448*x^9 + 101165629*x^8 - 500712216*x^7 - 2314815140*x^6 + 8765075664*x^5 + 30570952056*x^4 - 76367046112*x^3 - 215062276912*x^2 + 254794977984*x + 624192924176)
 
gp: K = bnfinit(x^16 - 8*x^15 - 308*x^14 + 2296*x^13 + 38866*x^12 - 263408*x^11 - 2607840*x^10 + 15509448*x^9 + 101165629*x^8 - 500712216*x^7 - 2314815140*x^6 + 8765075664*x^5 + 30570952056*x^4 - 76367046112*x^3 - 215062276912*x^2 + 254794977984*x + 624192924176, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 308 x^{14} + 2296 x^{13} + 38866 x^{12} - 263408 x^{11} - 2607840 x^{10} + 15509448 x^{9} + 101165629 x^{8} - 500712216 x^{7} - 2314815140 x^{6} + 8765075664 x^{5} + 30570952056 x^{4} - 76367046112 x^{3} - 215062276912 x^{2} + 254794977984 x + 624192924176 \)

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:  \(1186139225426340374118400000000000000=2^{32}\cdot 5^{14}\cdot 199^{2}\cdot 421^{2}\cdot 2539^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $179.74$
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:  $2, 5, 199, 421, 2539$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{104} a^{8} - \frac{1}{26} a^{7} - \frac{3}{52} a^{6} - \frac{5}{26} a^{5} + \frac{17}{104} a^{4} - \frac{5}{13} a^{3} - \frac{11}{26} a^{2} - \frac{1}{13} a + \frac{1}{26}$, $\frac{1}{104} a^{9} + \frac{1}{26} a^{7} + \frac{1}{13} a^{6} - \frac{11}{104} a^{5} - \frac{3}{13} a^{4} - \frac{11}{52} a^{3} + \frac{3}{13} a^{2} + \frac{3}{13} a + \frac{2}{13}$, $\frac{1}{104} a^{10} - \frac{1}{52} a^{7} - \frac{1}{8} a^{6} - \frac{11}{52} a^{5} - \frac{3}{26} a^{4} + \frac{7}{26} a^{3} - \frac{1}{13} a^{2} + \frac{6}{13} a - \frac{2}{13}$, $\frac{1}{104} a^{11} + \frac{5}{104} a^{7} - \frac{1}{13} a^{6} - \frac{1}{4} a^{5} - \frac{2}{13} a^{4} + \frac{2}{13} a^{3} - \frac{5}{13} a^{2} - \frac{4}{13} a + \frac{1}{13}$, $\frac{1}{208} a^{12} - \frac{1}{208} a^{10} - \frac{1}{208} a^{8} - \frac{1}{26} a^{7} - \frac{3}{208} a^{6} - \frac{1}{52} a^{5} - \frac{3}{13} a^{4} - \frac{9}{52} a^{3} + \frac{21}{52} a^{2} + \frac{1}{26} a + \frac{6}{13}$, $\frac{1}{208} a^{13} - \frac{1}{208} a^{11} - \frac{1}{208} a^{9} + \frac{17}{208} a^{7} - \frac{1}{4} a^{5} + \frac{3}{13} a^{4} + \frac{19}{52} a^{3} - \frac{2}{13} a^{2} + \frac{2}{13} a + \frac{2}{13}$, $\frac{1}{183299957170969278423745760864} a^{14} - \frac{7}{183299957170969278423745760864} a^{13} - \frac{335256455625337153667643965}{183299957170969278423745760864} a^{12} + \frac{249039145569626014085231565}{183299957170969278423745760864} a^{11} + \frac{139221135527240661345258299}{183299957170969278423745760864} a^{10} + \frac{19406517921278402206172685}{14099996705459175263365058528} a^{9} - \frac{98644250768828525955308891}{183299957170969278423745760864} a^{8} + \frac{22770294264670247781724901427}{183299957170969278423745760864} a^{7} - \frac{1332001035091414376971539807}{11456247323185579901484110054} a^{6} + \frac{2119464984960709715770007157}{91649978585484639211872880432} a^{5} + \frac{10834487148944610978109946693}{45824989292742319605936440216} a^{4} - \frac{21718196631689722609433243035}{45824989292742319605936440216} a^{3} + \frac{4397295242769935920891589747}{11456247323185579901484110054} a^{2} + \frac{363213899934124910592740205}{1762499588182396907920632316} a + \frac{1214849841667426637518548024}{5728123661592789950742055027}$, $\frac{1}{183299957170969278423745760864} a^{15} - \frac{12894479062512967448755539}{7049998352729587631682529264} a^{13} - \frac{167628227812668576833821937}{91649978585484639211872880432} a^{12} + \frac{59997783166112926010623469}{91649978585484639211872880432} a^{11} - \frac{267833453257546524911789659}{91649978585484639211872880432} a^{10} - \frac{11893838514361354139278359}{22912494646371159802968220108} a^{9} - \frac{416355068541355851465240459}{91649978585484639211872880432} a^{8} + \frac{9417573353914130162323514009}{183299957170969278423745760864} a^{7} + \frac{11246137458505357731593812975}{91649978585484639211872880432} a^{6} - \frac{20776007423313709540810606785}{91649978585484639211872880432} a^{5} - \frac{2460261396593871089002399385}{22912494646371159802968220108} a^{4} - \frac{16350723042527721751783977085}{45824989292742319605936440216} a^{3} + \frac{9883927276086025676727645011}{22912494646371159802968220108} a^{2} + \frac{5305621879300730617481852905}{22912494646371159802968220108} a - \frac{1630423740376795757913799649}{5728123661592789950742055027}$

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:  \( 8028159118800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1162:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1162
Character table for t16n1162 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.5120000000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$2$2.8.16.35$x^{8} + 6 x^{6} + 4 x^{5} + 10 x^{4} + 4 x^{2} + 12$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
2.8.16.35$x^{8} + 6 x^{6} + 4 x^{5} + 10 x^{4} + 4 x^{2} + 12$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
199Data not computed
421Data not computed
2539Data not computed