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Properties

Label 16.0.10386864596...0064.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 83^{8}$
Root discriminant $133.67$
Ramified primes $2, 83$
Class number $1638375$ (GRH)
Class group $[5, 5, 65535]$ (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![76059908671, -22375524712, 25396479484, -6286165360, 3766659562, -784977032, 324191712, -56416264, 17702027, -2519560, 627024, -69960, 14026, -1120, 180, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 180*x^14 - 1120*x^13 + 14026*x^12 - 69960*x^11 + 627024*x^10 - 2519560*x^9 + 17702027*x^8 - 56416264*x^7 + 324191712*x^6 - 784977032*x^5 + 3766659562*x^4 - 6286165360*x^3 + 25396479484*x^2 - 22375524712*x + 76059908671)
 
gp: K = bnfinit(x^16 - 8*x^15 + 180*x^14 - 1120*x^13 + 14026*x^12 - 69960*x^11 + 627024*x^10 - 2519560*x^9 + 17702027*x^8 - 56416264*x^7 + 324191712*x^6 - 784977032*x^5 + 3766659562*x^4 - 6286165360*x^3 + 25396479484*x^2 - 22375524712*x + 76059908671, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 180 x^{14} - 1120 x^{13} + 14026 x^{12} - 69960 x^{11} + 627024 x^{10} - 2519560 x^{9} + 17702027 x^{8} - 56416264 x^{7} + 324191712 x^{6} - 784977032 x^{5} + 3766659562 x^{4} - 6286165360 x^{3} + 25396479484 x^{2} - 22375524712 x + 76059908671 \)

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:  \(10386864596368228067918124169560064=2^{62}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.67$
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:  $2, 83$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2656=2^{5}\cdot 83\)
Dirichlet character group:    $\lbrace$$\chi_{2656}(1,·)$, $\chi_{2656}(1161,·)$, $\chi_{2656}(333,·)$, $\chi_{2656}(1493,·)$, $\chi_{2656}(665,·)$, $\chi_{2656}(165,·)$, $\chi_{2656}(1825,·)$, $\chi_{2656}(997,·)$, $\chi_{2656}(497,·)$, $\chi_{2656}(2157,·)$, $\chi_{2656}(829,·)$, $\chi_{2656}(1329,·)$, $\chi_{2656}(1993,·)$, $\chi_{2656}(2489,·)$, $\chi_{2656}(1661,·)$, $\chi_{2656}(2325,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{185727418680232864457521} a^{14} - \frac{7}{185727418680232864457521} a^{13} + \frac{73081017134257273656763}{185727418680232864457521} a^{12} - \frac{67031265445077913025445}{185727418680232864457521} a^{11} - \frac{48250134454982005315301}{185727418680232864457521} a^{10} - \frac{11024014986295804825514}{185727418680232864457521} a^{9} + \frac{6971178574563580355072}{185727418680232864457521} a^{8} + \frac{43825130444594922083271}{185727418680232864457521} a^{7} + \frac{57117857195318148097491}{185727418680232864457521} a^{6} - \frac{79444276021180883269536}{185727418680232864457521} a^{5} - \frac{52000710089310371696378}{185727418680232864457521} a^{4} + \frac{2823075317357000769538}{185727418680232864457521} a^{3} + \frac{61143593469905268938919}{185727418680232864457521} a^{2} + \frac{12788548860850784231126}{185727418680232864457521} a + \frac{2375502578885176437807}{5991207054201060143791}$, $\frac{1}{12637913024804179895882736172769} a^{15} + \frac{34022737}{12637913024804179895882736172769} a^{14} + \frac{4895427147764186045435024578243}{12637913024804179895882736172769} a^{13} - \frac{4058033637099892084673107201749}{12637913024804179895882736172769} a^{12} - \frac{3034996434605165745301092991575}{12637913024804179895882736172769} a^{11} + \frac{4988079379077661796043269779502}{12637913024804179895882736172769} a^{10} - \frac{910512880355184641369624165109}{12637913024804179895882736172769} a^{9} + \frac{6074032954617539520415427105192}{12637913024804179895882736172769} a^{8} + \frac{4717640577690859420678501495109}{12637913024804179895882736172769} a^{7} + \frac{2142312339946786123428915987184}{12637913024804179895882736172769} a^{6} - \frac{163295866820576730669184741439}{407674613703360641802668908799} a^{5} - \frac{661433055461638152096727241900}{12637913024804179895882736172769} a^{4} + \frac{5345395102040271237657473089143}{12637913024804179895882736172769} a^{3} + \frac{3538216176099935615011651220451}{12637913024804179895882736172769} a^{2} - \frac{925799347246927293819377648849}{12637913024804179895882736172769} a - \frac{194149148038701018036260057459}{407674613703360641802668908799}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{65535}$, which has order $1638375$ (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

\(\Q(\sqrt{-83}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-166}) \), \(\Q(\sqrt{2}, \sqrt{-83})\), \(\Q(\zeta_{16})^+\), 4.0.14108672.2, 8.0.199054625603584.9, \(\Q(\zeta_{32})^+\), 8.0.101915968309035008.12

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
83Data not computed