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Properties

Label 16.0.16351318827...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 101^{4}\cdot 991^{2}$
Root discriminant $50.22$
Ramified primes $2, 5, 101, 991$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28550621, -43770752, 43412784, -29495626, 16013087, -6811142, 2544433, -842176, 275476, -79934, 21258, -4478, 952, -194, 49, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 49*x^14 - 194*x^13 + 952*x^12 - 4478*x^11 + 21258*x^10 - 79934*x^9 + 275476*x^8 - 842176*x^7 + 2544433*x^6 - 6811142*x^5 + 16013087*x^4 - 29495626*x^3 + 43412784*x^2 - 43770752*x + 28550621)
 
gp: K = bnfinit(x^16 - 8*x^15 + 49*x^14 - 194*x^13 + 952*x^12 - 4478*x^11 + 21258*x^10 - 79934*x^9 + 275476*x^8 - 842176*x^7 + 2544433*x^6 - 6811142*x^5 + 16013087*x^4 - 29495626*x^3 + 43412784*x^2 - 43770752*x + 28550621, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 49 x^{14} - 194 x^{13} + 952 x^{12} - 4478 x^{11} + 21258 x^{10} - 79934 x^{9} + 275476 x^{8} - 842176 x^{7} + 2544433 x^{6} - 6811142 x^{5} + 16013087 x^{4} - 29495626 x^{3} + 43412784 x^{2} - 43770752 x + 28550621 \)

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:  \(1635131882791696000000000000=2^{16}\cdot 5^{12}\cdot 101^{4}\cdot 991^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.22$
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:  $2, 5, 101, 991$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{55} a^{12} + \frac{4}{11} a^{11} - \frac{5}{11} a^{10} + \frac{13}{55} a^{9} + \frac{1}{11} a^{7} + \frac{24}{55} a^{6} + \frac{4}{11} a^{5} + \frac{4}{11} a^{4} + \frac{2}{55} a^{3} - \frac{5}{11} a^{2} + \frac{3}{11} a + \frac{1}{5}$, $\frac{1}{55} a^{13} + \frac{3}{11} a^{11} + \frac{18}{55} a^{10} + \frac{3}{11} a^{9} + \frac{1}{11} a^{8} - \frac{21}{55} a^{7} - \frac{4}{11} a^{6} + \frac{1}{11} a^{5} - \frac{13}{55} a^{4} - \frac{2}{11} a^{3} + \frac{4}{11} a^{2} - \frac{14}{55} a$, $\frac{1}{55} a^{14} - \frac{7}{55} a^{11} + \frac{1}{11} a^{10} - \frac{5}{11} a^{9} - \frac{21}{55} a^{8} + \frac{3}{11} a^{7} - \frac{5}{11} a^{6} + \frac{17}{55} a^{5} + \frac{4}{11} a^{4} - \frac{2}{11} a^{3} - \frac{24}{55} a^{2} - \frac{1}{11} a$, $\frac{1}{22554092843797620057196729443074840003569795} a^{15} - \frac{2832903083682818685200898712824034444392}{369739226947501968150766056443849836124095} a^{14} + \frac{12644234655591396080799043722140548799424}{2050372076708874550654248131188621818506345} a^{13} + \frac{35215459045493402074039067497910177429678}{22554092843797620057196729443074840003569795} a^{12} + \frac{239442034468890639406840663390784193827989}{2050372076708874550654248131188621818506345} a^{11} + \frac{2473354729558734594002380790895746458695707}{22554092843797620057196729443074840003569795} a^{10} - \frac{3045697631279380018070096322162738428453426}{22554092843797620057196729443074840003569795} a^{9} + \frac{1007614207681488509015952248268478078900022}{2050372076708874550654248131188621818506345} a^{8} + \frac{4149129064655418286006931208769513309397501}{22554092843797620057196729443074840003569795} a^{7} + \frac{808348525068396721317112144815595519727507}{22554092843797620057196729443074840003569795} a^{6} - \frac{4708675739789180769116822600129391843386919}{22554092843797620057196729443074840003569795} a^{5} + \frac{4663360366424519931173947960549584859283418}{22554092843797620057196729443074840003569795} a^{4} - \frac{5180934025785936943777947632976701840565559}{22554092843797620057196729443074840003569795} a^{3} + \frac{9446751270731383322493951099879990274758328}{22554092843797620057196729443074840003569795} a^{2} - \frac{5612645271771328865246864116093547599079471}{22554092843797620057196729443074840003569795} a + \frac{155494289582982700031987252172433721577551}{410074415341774910130849626237724363701269}$

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:  \( -\frac{32192717138676196738058222441562}{1379033497022171816398454872704056252129} a^{15} + \frac{231167047960860048771022492410824}{113035532542800968557250399401971823945} a^{14} - \frac{1016397713684129540232469275653056}{125366681547470165127132261154914204739} a^{13} + \frac{343686197037723365343068105882130204}{6895167485110859081992274363520281260645} a^{12} - \frac{80294900542083895611504742110322618}{626833407737350825635661305774571023695} a^{11} + \frac{1393868488274552862660670945427268135}{1379033497022171816398454872704056252129} a^{10} - \frac{27316331686041551720355433741514311358}{6895167485110859081992274363520281260645} a^{9} + \frac{11536371747272811467048843354354764591}{626833407737350825635661305774571023695} a^{8} - \frac{73560032144447731989136189592761931990}{1379033497022171816398454872704056252129} a^{7} + \frac{1238600065470196201049995803831259774431}{6895167485110859081992274363520281260645} a^{6} - \frac{3447492177570042337054884449017437012692}{6895167485110859081992274363520281260645} a^{5} + \frac{2042494951606652787590115898504250723095}{1379033497022171816398454872704056252129} a^{4} - \frac{22131526292419027109646913097895791649792}{6895167485110859081992274363520281260645} a^{3} + \frac{38521556656581094518656552155382954355409}{6895167485110859081992274363520281260645} a^{2} - \frac{8942341517858297515497318606705700718834}{1379033497022171816398454872704056252129} a + \frac{3093353919162266864218275274801359133489}{626833407737350825635661305774571023695} \) (order $10$)
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:  \( 35334362.3413 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.404000000.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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}$
101Data not computed
991Data not computed