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Properties

Label 16.0.10873057608...1969.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 71^{8}$
Root discriminant $100.52$
Ramified primes $17, 71$
Class number $472311$ (GRH)
Class group $[21, 21, 1071]$ (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![22435022501, -5534375147, 8231592580, -1785014155, 1366555346, -257288560, 133700199, -21472404, 8415948, -1121047, 348512, -36670, 9259, -698, 144, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 144*x^14 - 698*x^13 + 9259*x^12 - 36670*x^11 + 348512*x^10 - 1121047*x^9 + 8415948*x^8 - 21472404*x^7 + 133700199*x^6 - 257288560*x^5 + 1366555346*x^4 - 1785014155*x^3 + 8231592580*x^2 - 5534375147*x + 22435022501)
 
gp: K = bnfinit(x^16 - 6*x^15 + 144*x^14 - 698*x^13 + 9259*x^12 - 36670*x^11 + 348512*x^10 - 1121047*x^9 + 8415948*x^8 - 21472404*x^7 + 133700199*x^6 - 257288560*x^5 + 1366555346*x^4 - 1785014155*x^3 + 8231592580*x^2 - 5534375147*x + 22435022501, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 144 x^{14} - 698 x^{13} + 9259 x^{12} - 36670 x^{11} + 348512 x^{10} - 1121047 x^{9} + 8415948 x^{8} - 21472404 x^{7} + 133700199 x^{6} - 257288560 x^{5} + 1366555346 x^{4} - 1785014155 x^{3} + 8231592580 x^{2} - 5534375147 x + 22435022501 \)

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:  \(108730576084219434179495230711969=17^{14}\cdot 71^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.52$
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:  $17, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1207=17\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{1207}(1,·)$, $\chi_{1207}(70,·)$, $\chi_{1207}(72,·)$, $\chi_{1207}(780,·)$, $\chi_{1207}(851,·)$, $\chi_{1207}(212,·)$, $\chi_{1207}(922,·)$, $\chi_{1207}(285,·)$, $\chi_{1207}(995,·)$, $\chi_{1207}(356,·)$, $\chi_{1207}(427,·)$, $\chi_{1207}(1135,·)$, $\chi_{1207}(1137,·)$, $\chi_{1207}(1206,·)$, $\chi_{1207}(569,·)$, $\chi_{1207}(638,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{26} a^{14} + \frac{3}{13} a^{12} + \frac{1}{26} a^{11} + \frac{4}{13} a^{9} - \frac{9}{26} a^{8} - \frac{3}{13} a^{7} - \frac{4}{13} a^{6} - \frac{5}{26} a^{5} + \frac{2}{13} a^{4} - \frac{5}{13} a^{3} - \frac{1}{26} a^{2} - \frac{3}{13} a - \frac{6}{13}$, $\frac{1}{321139586772639059881889258816317852019051718896566} a^{15} - \frac{5265659917951069698655263688813970908871571856335}{321139586772639059881889258816317852019051718896566} a^{14} + \frac{46517503818797734285599045208932432172453780778439}{321139586772639059881889258816317852019051718896566} a^{13} + \frac{70687807120625351989378609822072780540890220448007}{321139586772639059881889258816317852019051718896566} a^{12} + \frac{611352725820979683084674518358171491162112356767}{3179599869036030295860289691250671802168828899966} a^{11} - \frac{124719334774297514485499881442855463604080016309469}{321139586772639059881889258816317852019051718896566} a^{10} + \frac{144315443496596655349600823907615544806620184128597}{321139586772639059881889258816317852019051718896566} a^{9} + \frac{84571927098409897609216065469992540094364475641357}{321139586772639059881889258816317852019051718896566} a^{8} - \frac{84394829116743026114112555827597836954567364849335}{321139586772639059881889258816317852019051718896566} a^{7} + \frac{19951829912975957226946278871646859001048618649215}{321139586772639059881889258816317852019051718896566} a^{6} + \frac{9208426893138936107243927794066949824678032036797}{321139586772639059881889258816317852019051718896566} a^{5} - \frac{138994652090436472715197085698765796921963381656795}{321139586772639059881889258816317852019051718896566} a^{4} - \frac{57590261911851162224134937960803934943909807968955}{321139586772639059881889258816317852019051718896566} a^{3} - \frac{51305370300454553337438559171149627818687216566007}{321139586772639059881889258816317852019051718896566} a^{2} - \frac{28666756133199335957890473950352219817380397605219}{321139586772639059881889258816317852019051718896566} a + \frac{33177975361835755749237587780526688725209989505}{67437964462964943276331217727072207479851263943}$

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

Class group and class number

$C_{21}\times C_{21}\times C_{1071}$, which has order $472311$ (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:  \( 3640.01221338 \) (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

\(\Q(\sqrt{-1207}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-71}) \), \(\Q(\sqrt{17}, \sqrt{-71})\), 4.4.4913.1, 4.0.24766433.1, 8.0.613376203543489.1, 8.0.10427395460239313.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
71Data not computed