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Properties

Label 16.0.21686570067...1984.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 7^{8}\cdot 13^{8}$
Root discriminant $139.96$
Ramified primes $2, 7, 13$
Class number $2950656$ (GRH)
Class group $[16, 184416]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![148858281023, -40815400920, 45932378476, -10616762656, 6284093722, -1223807560, 498035776, -80960632, 24998651, -3318856, 812720, -84344, 16666, -1232, 196, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 196*x^14 - 1232*x^13 + 16666*x^12 - 84344*x^11 + 812720*x^10 - 3318856*x^9 + 24998651*x^8 - 80960632*x^7 + 498035776*x^6 - 1223807560*x^5 + 6284093722*x^4 - 10616762656*x^3 + 45932378476*x^2 - 40815400920*x + 148858281023)
 
gp: K = bnfinit(x^16 - 8*x^15 + 196*x^14 - 1232*x^13 + 16666*x^12 - 84344*x^11 + 812720*x^10 - 3318856*x^9 + 24998651*x^8 - 80960632*x^7 + 498035776*x^6 - 1223807560*x^5 + 6284093722*x^4 - 10616762656*x^3 + 45932378476*x^2 - 40815400920*x + 148858281023, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 196 x^{14} - 1232 x^{13} + 16666 x^{12} - 84344 x^{11} + 812720 x^{10} - 3318856 x^{9} + 24998651 x^{8} - 80960632 x^{7} + 498035776 x^{6} - 1223807560 x^{5} + 6284093722 x^{4} - 10616762656 x^{3} + 45932378476 x^{2} - 40815400920 x + 148858281023 \)

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:  \(21686570067329360666414797746601984=2^{62}\cdot 7^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.96$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2912=2^{5}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2912}(1,·)$, $\chi_{2912}(1093,·)$, $\chi_{2912}(2185,·)$, $\chi_{2912}(909,·)$, $\chi_{2912}(2001,·)$, $\chi_{2912}(729,·)$, $\chi_{2912}(1821,·)$, $\chi_{2912}(545,·)$, $\chi_{2912}(1637,·)$, $\chi_{2912}(2729,·)$, $\chi_{2912}(365,·)$, $\chi_{2912}(1457,·)$, $\chi_{2912}(2549,·)$, $\chi_{2912}(1273,·)$, $\chi_{2912}(2365,·)$, $\chi_{2912}(181,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{17} a^{11} + \frac{3}{17} a^{10} - \frac{3}{17} a^{9} + \frac{8}{17} a^{8} - \frac{5}{17} a^{7} + \frac{4}{17} a^{6} - \frac{2}{17} a^{5} - \frac{7}{17} a^{4} - \frac{6}{17} a^{2} - \frac{2}{17} a - \frac{4}{17}$, $\frac{1}{17} a^{12} + \frac{5}{17} a^{10} + \frac{5}{17} a^{8} + \frac{2}{17} a^{7} + \frac{3}{17} a^{6} - \frac{1}{17} a^{5} + \frac{4}{17} a^{4} - \frac{6}{17} a^{3} - \frac{1}{17} a^{2} + \frac{2}{17} a - \frac{5}{17}$, $\frac{1}{17} a^{13} + \frac{2}{17} a^{10} + \frac{3}{17} a^{9} - \frac{4}{17} a^{8} - \frac{6}{17} a^{7} - \frac{4}{17} a^{6} - \frac{3}{17} a^{5} - \frac{5}{17} a^{4} - \frac{1}{17} a^{3} - \frac{2}{17} a^{2} + \frac{5}{17} a + \frac{3}{17}$, $\frac{1}{31458164344899280495681} a^{14} - \frac{7}{31458164344899280495681} a^{13} + \frac{418673743923955632756}{31458164344899280495681} a^{12} - \frac{661562207961423179052}{31458164344899280495681} a^{11} + \frac{11764934669801731928463}{31458164344899280495681} a^{10} - \frac{13591854366203372433020}{31458164344899280495681} a^{9} - \frac{4043434904822007719662}{31458164344899280495681} a^{8} - \frac{7627772016929852214347}{31458164344899280495681} a^{7} + \frac{5875583698713093754402}{31458164344899280495681} a^{6} + \frac{4783661199961533425464}{31458164344899280495681} a^{5} - \frac{15600108525665956145907}{31458164344899280495681} a^{4} - \frac{9930575648946244568611}{31458164344899280495681} a^{3} - \frac{4122769224520214847350}{31458164344899280495681} a^{2} - \frac{15377263062491319685348}{31458164344899280495681} a + \frac{7331601226767268745214}{31458164344899280495681}$, $\frac{1}{176596602808468056220762942657} a^{15} + \frac{2806841}{176596602808468056220762942657} a^{14} - \frac{4090117940720092815998380473}{176596602808468056220762942657} a^{13} + \frac{1790831736498816370911411245}{176596602808468056220762942657} a^{12} + \frac{2447686973257113703393307395}{176596602808468056220762942657} a^{11} - \frac{15604385707239926121638224419}{176596602808468056220762942657} a^{10} + \frac{14045009366658465260154300570}{176596602808468056220762942657} a^{9} + \frac{71497209924830516480819272496}{176596602808468056220762942657} a^{8} - \frac{4162245343109358695855402256}{176596602808468056220762942657} a^{7} - \frac{17441413877368794980570775934}{176596602808468056220762942657} a^{6} - \frac{7165153713399359208729907221}{176596602808468056220762942657} a^{5} - \frac{83288860535712708806577953781}{176596602808468056220762942657} a^{4} + \frac{790945158039892327849515196}{10388035459321650365927231921} a^{3} + \frac{10596387948089030925847488}{335097918042633882771846191} a^{2} - \frac{32378149650886818764046801869}{176596602808468056220762942657} a - \frac{2297551663235921991231480244}{5696664606724776007121385247}$

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

Class group and class number

$C_{16}\times C_{184416}$, which has order $2950656$ (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:  \( -1 \) (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:  \( 15753.94986242651 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-182}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-91}) \), \(\Q(\sqrt{2}, \sqrt{-91})\), \(\Q(\zeta_{16})^+\), 4.0.16959488.5, 8.0.287624233222144.127, 8.0.147263607409737728.1, \(\Q(\zeta_{32})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed