Properties

Label 16.0.3243658447265625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{14}$
Root discriminant $9.32$
Ramified primes $3, 5$
Class number $1$
Class group Trivial
Galois Group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 16, -30, 57, -110, 183, -240, 250, -215, 157, -100, 57, -30, 14, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 30*x^13 + 57*x^12 - 100*x^11 + 157*x^10 - 215*x^9 + 250*x^8 - 240*x^7 + 183*x^6 - 110*x^5 + 57*x^4 - 30*x^3 + 16*x^2 - 5*x + 1)
gp: K = bnfinit(x^16 - 5*x^15 + 14*x^14 - 30*x^13 + 57*x^12 - 100*x^11 + 157*x^10 - 215*x^9 + 250*x^8 - 240*x^7 + 183*x^6 - 110*x^5 + 57*x^4 - 30*x^3 + 16*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 5 x^{15} \) \(\mathstrut +\mathstrut 14 x^{14} \) \(\mathstrut -\mathstrut 30 x^{13} \) \(\mathstrut +\mathstrut 57 x^{12} \) \(\mathstrut -\mathstrut 100 x^{11} \) \(\mathstrut +\mathstrut 157 x^{10} \) \(\mathstrut -\mathstrut 215 x^{9} \) \(\mathstrut +\mathstrut 250 x^{8} \) \(\mathstrut -\mathstrut 240 x^{7} \) \(\mathstrut +\mathstrut 183 x^{6} \) \(\mathstrut -\mathstrut 110 x^{5} \) \(\mathstrut +\mathstrut 57 x^{4} \) \(\mathstrut -\mathstrut 30 x^{3} \) \(\mathstrut +\mathstrut 16 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(3243658447265625=3^{12}\cdot 5^{14}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.32$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{487621} a^{15} + \frac{220676}{487621} a^{14} - \frac{196521}{487621} a^{13} + \frac{73288}{487621} a^{12} - \frac{144143}{487621} a^{11} - \frac{153169}{487621} a^{10} - \frac{87833}{487621} a^{9} - \frac{139738}{487621} a^{8} + \frac{118333}{487621} a^{7} - \frac{210501}{487621} a^{6} + \frac{131188}{487621} a^{5} + \frac{152527}{487621} a^{4} - \frac{179065}{487621} a^{3} + \frac{74924}{487621} a^{2} + \frac{50392}{487621} a - \frac{127579}{487621}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{391279}{487621} a^{15} - \frac{1554455}{487621} a^{14} + \frac{3778962}{487621} a^{13} - \frac{7274731}{487621} a^{12} + \frac{13232214}{487621} a^{11} - \frac{22209470}{487621} a^{10} + \frac{32502659}{487621} a^{9} - \frac{40074715}{487621} a^{8} + \frac{40126016}{487621} a^{7} - \frac{30790171}{487621} a^{6} + \frac{15825896}{487621} a^{5} - \frac{5661230}{487621} a^{4} + \frac{2862597}{487621} a^{3} - \frac{3000071}{487621} a^{2} + \frac{863854}{487621} a + \frac{241092}{487621} \) (order $30$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{218084}{487621} a^{15} - \frac{1312674}{487621} a^{14} + \frac{3800136}{487621} a^{13} - \frac{8102882}{487621} a^{12} + \frac{15297246}{487621} a^{11} - \frac{27025988}{487621} a^{10} + \frac{42666798}{487621} a^{9} - \frac{58286875}{487621} a^{8} + \frac{67459487}{487621} a^{7} - \frac{63699390}{487621} a^{6} + \frac{46628475}{487621} a^{5} - \frac{25699781}{487621} a^{4} + \frac{12594471}{487621} a^{3} - \frac{6793167}{487621} a^{2} + \frac{3100177}{487621} a - \frac{747239}{487621} \),  \( \frac{26168}{487621} a^{15} - \frac{245935}{487621} a^{14} + \frac{864780}{487621} a^{13} - \frac{1963493}{487621} a^{12} + \frac{3715379}{487621} a^{11} - \frac{6708466}{487621} a^{10} + \frac{10959112}{487621} a^{9} - \frac{15597977}{487621} a^{8} + \frac{18674192}{487621} a^{7} - \frac{18265329}{487621} a^{6} + \frac{13729132}{487621} a^{5} - \frac{7165664}{487621} a^{4} + \frac{2702995}{487621} a^{3} - \frac{1575672}{487621} a^{2} + \frac{1105914}{487621} a - \frac{233906}{487621} \),  \( \frac{53423}{487621} a^{15} - \frac{38969}{487621} a^{14} - \frac{261253}{487621} a^{13} + \frac{1131057}{487621} a^{12} - \frac{2478762}{487621} a^{11} + \frac{4896724}{487621} a^{10} - \frac{9190275}{487621} a^{9} + \frac{15370587}{487621} a^{8} - \frac{21757730}{487621} a^{7} + \frac{25276871}{487621} a^{6} - \frac{23525917}{487621} a^{5} + \frac{15419262}{487621} a^{4} - \frac{6867411}{487621} a^{3} + \frac{2222168}{487621} a^{2} - \frac{2014209}{487621} a + \frac{801042}{487621} \),  \( \frac{76028}{487621} a^{15} - \frac{508440}{487621} a^{14} + \frac{1559336}{487621} a^{13} - \frac{3518450}{487621} a^{12} + \frac{6717044}{487621} a^{11} - \frac{11958535}{487621} a^{10} + \frac{19219490}{487621} a^{9} - \frac{27021092}{487621} a^{8} + \frac{32196860}{487621} a^{7} - \frac{31456552}{487621} a^{6} + \frac{24054759}{487621} a^{5} - \frac{12958012}{487621} a^{4} + \frac{5300710}{487621} a^{3} - \frac{2017134}{487621} a^{2} + \frac{1427642}{487621} a - \frac{306901}{487621} \),  \( \frac{250041}{487621} a^{15} - \frac{1144644}{487621} a^{14} + \frac{3161777}{487621} a^{13} - \frac{6619066}{487621} a^{12} + \frac{12549256}{487621} a^{11} - \frac{21744292}{487621} a^{10} + \frac{33756915}{487621} a^{9} - \frac{45582877}{487621} a^{8} + \frac{51922441}{487621} a^{7} - \frac{48344280}{487621} a^{6} + \frac{35222750}{487621} a^{5} - \frac{19802506}{487621} a^{4} + \frac{9033354}{487621} a^{3} - \frac{4715525}{487621} a^{2} + \frac{2377537}{487621} a - \frac{790161}{487621} \),  \( \frac{322766}{487621} a^{15} - \frac{1552517}{487621} a^{14} + \frac{3831183}{487621} a^{13} - \frac{7422038}{487621} a^{12} + \frac{13601081}{487621} a^{11} - \frac{23208556}{487621} a^{10} + \frac{34424711}{487621} a^{9} - \frac{43081561}{487621} a^{8} + \frac{44352522}{487621} a^{7} - \frac{35002443}{487621} a^{6} + \frac{18986071}{487621} a^{5} - \frac{7000793}{487621} a^{4} + \frac{3573824}{487621} a^{3} - \frac{2594195}{487621} a^{2} + \frac{225817}{487621} a - \frac{32927}{487621} \),  \( \frac{505949}{487621} a^{15} - \frac{1729330}{487621} a^{14} + \frac{4120407}{487621} a^{13} - \frac{7975327}{487621} a^{12} + \frac{14706304}{487621} a^{11} - \frac{24428385}{487621} a^{10} + \frac{35430030}{487621} a^{9} - \frac{44018462}{487621} a^{8} + \frac{44730148}{487621} a^{7} - \frac{36088930}{487621} a^{6} + \frac{21409837}{487621} a^{5} - \frac{11231620}{487621} a^{4} + \frac{5637462}{487621} a^{3} - \frac{3834632}{487621} a^{2} + \frac{1493265}{487621} a - \frac{612838}{487621} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 112.874368042 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})\), 8.4.56953125.1 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
5Data not computed