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Properties

Label 16.0.11034809241...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 17^{14}$
Root discriminant $75.45$
Ramified primes $2, 5, 17$
Class number $31488$ (GRH)
Class group $[16, 1968]$ (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![375033361, -53576192, 190527196, -24484700, 45778750, -5234808, 6785394, -678478, 679239, -57752, 47186, -3254, 2237, -114, 67, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 67*x^14 - 114*x^13 + 2237*x^12 - 3254*x^11 + 47186*x^10 - 57752*x^9 + 679239*x^8 - 678478*x^7 + 6785394*x^6 - 5234808*x^5 + 45778750*x^4 - 24484700*x^3 + 190527196*x^2 - 53576192*x + 375033361)
 
gp: K = bnfinit(x^16 - 2*x^15 + 67*x^14 - 114*x^13 + 2237*x^12 - 3254*x^11 + 47186*x^10 - 57752*x^9 + 679239*x^8 - 678478*x^7 + 6785394*x^6 - 5234808*x^5 + 45778750*x^4 - 24484700*x^3 + 190527196*x^2 - 53576192*x + 375033361, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 67 x^{14} - 114 x^{13} + 2237 x^{12} - 3254 x^{11} + 47186 x^{10} - 57752 x^{9} + 679239 x^{8} - 678478 x^{7} + 6785394 x^{6} - 5234808 x^{5} + 45778750 x^{4} - 24484700 x^{3} + 190527196 x^{2} - 53576192 x + 375033361 \)

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:  \(1103480924139689928294400000000=2^{24}\cdot 5^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.45$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(680=2^{3}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(259,·)$, $\chi_{680}(321,·)$, $\chi_{680}(339,·)$, $\chi_{680}(81,·)$, $\chi_{680}(19,·)$, $\chi_{680}(659,·)$, $\chi_{680}(441,·)$, $\chi_{680}(281,·)$, $\chi_{680}(219,·)$, $\chi_{680}(161,·)$, $\chi_{680}(361,·)$, $\chi_{680}(579,·)$, $\chi_{680}(179,·)$, $\chi_{680}(121,·)$, $\chi_{680}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{103} a^{14} + \frac{33}{103} a^{13} - \frac{47}{103} a^{12} + \frac{36}{103} a^{11} + \frac{1}{103} a^{10} + \frac{22}{103} a^{9} + \frac{28}{103} a^{8} - \frac{24}{103} a^{7} + \frac{44}{103} a^{6} + \frac{49}{103} a^{5} + \frac{17}{103} a^{4} - \frac{31}{103} a^{3} - \frac{10}{103} a^{2} - \frac{20}{103}$, $\frac{1}{104802596454418265300917979602895750720245271} a^{15} + \frac{390039155090566331206038192619229045365641}{104802596454418265300917979602895750720245271} a^{14} - \frac{21549293127124439061932342685666093275154406}{104802596454418265300917979602895750720245271} a^{13} + \frac{22995375481953264193714004056877776827205165}{104802596454418265300917979602895750720245271} a^{12} + \frac{15437312646446445842806452926761132489439861}{104802596454418265300917979602895750720245271} a^{11} - \frac{34414866474346166336906428128755871331931905}{104802596454418265300917979602895750720245271} a^{10} + \frac{34953051869070067461172707410102264233748701}{104802596454418265300917979602895750720245271} a^{9} - \frac{40608361284876245415276713664371919818465198}{104802596454418265300917979602895750720245271} a^{8} + \frac{34619225511803398510647772075311206013620008}{104802596454418265300917979602895750720245271} a^{7} + \frac{18269979957173556280016949594452668832485066}{104802596454418265300917979602895750720245271} a^{6} - \frac{6494151554292642532839062769708882577604940}{104802596454418265300917979602895750720245271} a^{5} - \frac{5642997423852110120542357178958462741117957}{104802596454418265300917979602895750720245271} a^{4} - \frac{32088062164191867927474257166169103878575604}{104802596454418265300917979602895750720245271} a^{3} - \frac{113481061731066046889066035924664380067950}{104802596454418265300917979602895750720245271} a^{2} - \frac{37679955864086160068689282034659619604122336}{104802596454418265300917979602895750720245271} a + \frac{34998864345646828626682687928207457548253178}{104802596454418265300917979602895750720245271}$

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

Class group and class number

$C_{16}\times C_{1968}$, which has order $31488$ (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:  \( 3640.012213375973 \) (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

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-170}) \), \(\Q(\sqrt{-10}, \sqrt{17})\), 4.4.4913.1, 4.0.7860800.1, 8.0.61792176640000.44, 8.0.1050467002880000.6, \(\Q(\zeta_{17})^+\)

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ 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
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
17Data not computed