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Properties

Label 16.8.64557605582...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 13^{2}\cdot 97^{4}\cdot 101^{10}$
Root discriminant $173.03$
Ramified primes $5, 13, 97, 101$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group 16T860

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-406670525, -6979244050, -15689345750, -17661455410, -12573808799, -5169819990, -1020941616, -80124311, 12524048, 4108102, 560949, 28486, -6364, -803, -62, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 62*x^14 - 803*x^13 - 6364*x^12 + 28486*x^11 + 560949*x^10 + 4108102*x^9 + 12524048*x^8 - 80124311*x^7 - 1020941616*x^6 - 5169819990*x^5 - 12573808799*x^4 - 17661455410*x^3 - 15689345750*x^2 - 6979244050*x - 406670525)
 
gp: K = bnfinit(x^16 - x^15 - 62*x^14 - 803*x^13 - 6364*x^12 + 28486*x^11 + 560949*x^10 + 4108102*x^9 + 12524048*x^8 - 80124311*x^7 - 1020941616*x^6 - 5169819990*x^5 - 12573808799*x^4 - 17661455410*x^3 - 15689345750*x^2 - 6979244050*x - 406670525, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 62 x^{14} - 803 x^{13} - 6364 x^{12} + 28486 x^{11} + 560949 x^{10} + 4108102 x^{9} + 12524048 x^{8} - 80124311 x^{7} - 1020941616 x^{6} - 5169819990 x^{5} - 12573808799 x^{4} - 17661455410 x^{3} - 15689345750 x^{2} - 6979244050 x - 406670525 \)

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:  \(645576055826149774217152338862890625=5^{8}\cdot 13^{2}\cdot 97^{4}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $173.03$
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:  $5, 13, 97, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{10} + \frac{1}{10} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{3}{10} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{10} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{2}{5} a^{11} + \frac{1}{10} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{3}{10} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{10} a^{4} + \frac{2}{5} a^{3} - \frac{1}{2} a$, $\frac{1}{170} a^{14} - \frac{3}{170} a^{13} + \frac{1}{34} a^{12} - \frac{61}{170} a^{11} + \frac{47}{170} a^{10} + \frac{21}{170} a^{9} - \frac{13}{34} a^{8} - \frac{33}{170} a^{7} + \frac{23}{170} a^{6} - \frac{83}{170} a^{5} - \frac{3}{34} a^{4} + \frac{29}{170} a^{3} + \frac{69}{170} a^{2} - \frac{13}{34} a + \frac{7}{34}$, $\frac{1}{461079777295604283404463545279874347894429572050693855880652544348287904650} a^{15} + \frac{4125395724842561749516349631385779993890944884424087727571360300024571}{27122339840917899023791973251757314582025268944158462110626620255781641450} a^{14} + \frac{6376645276131651310390982026847007775224855329371145208413000626378804029}{461079777295604283404463545279874347894429572050693855880652544348287904650} a^{13} - \frac{22459832922598060312060671661117716651285341426472972195132006996958293591}{461079777295604283404463545279874347894429572050693855880652544348287904650} a^{12} + \frac{2622386982493310628790075468230008770654987805209899655958211484161532979}{27122339840917899023791973251757314582025268944158462110626620255781641450} a^{11} - \frac{831707815554212127311100807054093675427077059461166056326101216357289983}{8383268678101896061899337186906806325353264946376251925102773533605234630} a^{10} - \frac{168380025275787828573431834208281935363670748107677572663360834967337574651}{461079777295604283404463545279874347894429572050693855880652544348287904650} a^{9} + \frac{109030926295483558654069221671482648734544198154540455706671183962331496599}{461079777295604283404463545279874347894429572050693855880652544348287904650} a^{8} - \frac{16238244151850401122825794369651987632513028654732103029127326870839773289}{92215955459120856680892709055974869578885914410138771176130508869657580930} a^{7} + \frac{184662147998767565074042196168622514782361589254735570743463876802100830159}{461079777295604283404463545279874347894429572050693855880652544348287904650} a^{6} + \frac{118603340578052998113148943768235441046030007501937677944224763853671022821}{461079777295604283404463545279874347894429572050693855880652544348287904650} a^{5} - \frac{175912462783554470710579639898239216620057677311379250244683116536956803557}{461079777295604283404463545279874347894429572050693855880652544348287904650} a^{4} + \frac{14543237849167150618859121256546743867650097820448843938115430320663987549}{92215955459120856680892709055974869578885914410138771176130508869657580930} a^{3} - \frac{20844865568419027584078023378033003783145701004730102047319612931581170837}{92215955459120856680892709055974869578885914410138771176130508869657580930} a^{2} + \frac{4957308989142382940631708126470825837379226031721384747269376530169527783}{18443191091824171336178541811194973915777182882027754235226101773931516186} a + \frac{9753742453936669403008433583648498773042861300948217613459988290076891}{20538074712499077211780113375495516609996862897581018079316371685892557}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  $11$
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:  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:  \( 246829296389 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T860:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n860 are not computed
Character table for t16n860 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{505}) \), 4.4.2525.1 x2, 4.4.51005.1 x2, \(\Q(\sqrt{5}, \sqrt{101})\), 8.8.65037750625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
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}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
101Data not computed