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Properties

Label 16.8.19925537421...0896.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 3^{14}\cdot 13^{6}\cdot 29^{8}\cdot 107^{8}\cdot 139^{8}$
Root discriminant $50{,}840.89$
Ramified primes $2, 3, 13, 29, 107, 139$
Class number Not computed
Class group Not computed
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12477755900910335205401361, 0, 67888704460353159384144, 0, -617059300053864361194, 0, 135885537776647872, 0, 1804162994563848, 0, 68564686032, 0, -84015162, 0, -1680, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1680*x^14 - 84015162*x^12 + 68564686032*x^10 + 1804162994563848*x^8 + 135885537776647872*x^6 - 617059300053864361194*x^4 + 67888704460353159384144*x^2 + 12477755900910335205401361)
 
gp: K = bnfinit(x^16 - 1680*x^14 - 84015162*x^12 + 68564686032*x^10 + 1804162994563848*x^8 + 135885537776647872*x^6 - 617059300053864361194*x^4 + 67888704460353159384144*x^2 + 12477755900910335205401361, 1)
 

Normalized defining polynomial

\( x^{16} - 1680 x^{14} - 84015162 x^{12} + 68564686032 x^{10} + 1804162994563848 x^{8} + 135885537776647872 x^{6} - 617059300053864361194 x^{4} + 67888704460353159384144 x^{2} + 12477755900910335205401361 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1992553742188905474600208423051202739081327305267455857076982769849672400896=2^{56}\cdot 3^{14}\cdot 13^{6}\cdot 29^{8}\cdot 107^{8}\cdot 139^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50{,}840.89$
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:  $2, 3, 13, 29, 107, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{1224033} a^{12} - \frac{2893}{1224033} a^{10} - \frac{104170}{1224033} a^{8} + \frac{56362}{408011} a^{6} + \frac{175232}{408011} a^{4} + \frac{122047}{408011} a^{2} - \frac{80683}{408011}$, $\frac{1}{1224033} a^{13} - \frac{2893}{1224033} a^{11} - \frac{104170}{1224033} a^{9} + \frac{56362}{408011} a^{7} + \frac{175232}{408011} a^{5} + \frac{122047}{408011} a^{3} - \frac{80683}{408011} a$, $\frac{1}{701156655725811966685499413809463729044127137165988541556091121763495162717} a^{14} - \frac{21607711715185131358452339879295008712208646151528622806389400765889}{233718885241937322228499804603154576348042379055329513852030373921165054239} a^{12} + \frac{28656127395545972647562722983333004772487268806919766163743295736324791029}{233718885241937322228499804603154576348042379055329513852030373921165054239} a^{10} - \frac{33570255132030748093381111059503735152884801110552945440582858595518051684}{233718885241937322228499804603154576348042379055329513852030373921165054239} a^{8} - \frac{99216613472433519342378865846057805300046921955494329946499531237429842176}{233718885241937322228499804603154576348042379055329513852030373921165054239} a^{6} - \frac{15701103835637526500202343655714422699869184812983517183107096369900095426}{77906295080645774076166601534384858782680793018443171284010124640388351413} a^{4} + \frac{24002896630715012258349669893077148598248167598788083030953174392379961083}{77906295080645774076166601534384858782680793018443171284010124640388351413} a^{2} + \frac{7978562151137330948074163739253753751009980398858744874171577809961035}{5992791929280444159705123194952681444821599462957167021846932664645257801}$, $\frac{1}{63506575781600782623173465551413128197627751139125526180977467111752252696924434138157} a^{15} - \frac{1803767771503108055226502470664799708827849739167267891018458262041800563797298}{7056286197955642513685940616823680910847527904347280686775274123528028077436048237573} a^{13} + \frac{3058700049587142893081612310015560866807342358878423479269442526518982215698316976895}{21168858593866927541057821850471042732542583713041842060325822370584084232308144712719} a^{11} + \frac{3205789047289236363364251570012574210489510982614511851252679306384985251138586740069}{21168858593866927541057821850471042732542583713041842060325822370584084232308144712719} a^{9} + \frac{9874465040884880569266907215011879980753362568312389748672635626020628147076507133576}{21168858593866927541057821850471042732542583713041842060325822370584084232308144712719} a^{7} - \frac{1388895842907633177911314344744236026504746748743436455424730367998484567141746730983}{7056286197955642513685940616823680910847527904347280686775274123528028077436048237573} a^{5} - \frac{2221616061704001753912953153611014861134331634089178255638037228113898569521772477687}{7056286197955642513685940616823680910847527904347280686775274123528028077436048237573} a^{3} - \frac{251492858711066065263610748420645370348956709978649205745333232760965207558777774198}{542791245996587885668149278217206223911348300334406206675021086425232929033542172121} a$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.1393025246413632.1, 4.4.33432605913927168.1, 4.4.13824.1, 8.8.4470956552783831393152593634000896.2

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
3Data not computed
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$107$107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.8.4.1$x^{8} + 45796 x^{4} - 1225043 x^{2} + 524318404$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$139$139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$