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Properties

Label 16.0.3243658447265625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.244\times 10^{15}$
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: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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(y^16 - 5*y^15 + 14*y^14 - 30*y^13 + 57*y^12 - 100*y^11 + 157*y^10 - 215*y^9 + 250*y^8 - 240*y^7 + 183*y^6 - 110*y^5 + 57*y^4 - 30*y^3 + 16*y^2 - 5*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); 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);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(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)
 

\( x^{16} - 5 x^{15} + 14 x^{14} - 30 x^{13} + 57 x^{12} - 100 x^{11} + 157 x^{10} - 215 x^{9} + 250 x^{8} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3243658447265625\) \(\medspace = 3^{12}\cdot 5^{14}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(9.32\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{3/4}5^{7/8}\approx 9.320510388204081$
Ramified primes:   \(3\), \(5\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

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

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{587011}{487621} a^{15} + \frac{2643045}{487621} a^{14} - \frac{6751280}{487621} a^{13} + \frac{13641566}{487621} a^{12} - \frac{25288502}{487621} a^{11} + \frac{43337559}{487621} a^{10} - \frac{65498107}{487621} a^{9} + \frac{84984552}{487621} a^{8} - \frac{91858719}{487621} a^{7} + \frac{79309987}{487621} a^{6} - \frac{51477606}{487621} a^{5} + \frac{25347031}{487621} a^{4} - \frac{12111433}{487621} a^{3} + \frac{7078246}{487621} a^{2} - \frac{3518936}{487621} a + \frac{467947}{487621} \)  (order $30$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{218084}{487621}a^{15}-\frac{1312674}{487621}a^{14}+\frac{3800136}{487621}a^{13}-\frac{8102882}{487621}a^{12}+\frac{15297246}{487621}a^{11}-\frac{27025988}{487621}a^{10}+\frac{42666798}{487621}a^{9}-\frac{58286875}{487621}a^{8}+\frac{67459487}{487621}a^{7}-\frac{63699390}{487621}a^{6}+\frac{46628475}{487621}a^{5}-\frac{25699781}{487621}a^{4}+\frac{12594471}{487621}a^{3}-\frac{6793167}{487621}a^{2}+\frac{3100177}{487621}a-\frac{747239}{487621}$, $\frac{26168}{487621}a^{15}-\frac{245935}{487621}a^{14}+\frac{864780}{487621}a^{13}-\frac{1963493}{487621}a^{12}+\frac{3715379}{487621}a^{11}-\frac{6708466}{487621}a^{10}+\frac{10959112}{487621}a^{9}-\frac{15597977}{487621}a^{8}+\frac{18674192}{487621}a^{7}-\frac{18265329}{487621}a^{6}+\frac{13729132}{487621}a^{5}-\frac{7165664}{487621}a^{4}+\frac{2702995}{487621}a^{3}-\frac{1575672}{487621}a^{2}+\frac{1105914}{487621}a-\frac{233906}{487621}$, $\frac{53423}{487621}a^{15}-\frac{38969}{487621}a^{14}-\frac{261253}{487621}a^{13}+\frac{1131057}{487621}a^{12}-\frac{2478762}{487621}a^{11}+\frac{4896724}{487621}a^{10}-\frac{9190275}{487621}a^{9}+\frac{15370587}{487621}a^{8}-\frac{21757730}{487621}a^{7}+\frac{25276871}{487621}a^{6}-\frac{23525917}{487621}a^{5}+\frac{15419262}{487621}a^{4}-\frac{6867411}{487621}a^{3}+\frac{2222168}{487621}a^{2}-\frac{2014209}{487621}a+\frac{801042}{487621}$, $\frac{76028}{487621}a^{15}-\frac{508440}{487621}a^{14}+\frac{1559336}{487621}a^{13}-\frac{3518450}{487621}a^{12}+\frac{6717044}{487621}a^{11}-\frac{11958535}{487621}a^{10}+\frac{19219490}{487621}a^{9}-\frac{27021092}{487621}a^{8}+\frac{32196860}{487621}a^{7}-\frac{31456552}{487621}a^{6}+\frac{24054759}{487621}a^{5}-\frac{12958012}{487621}a^{4}+\frac{5300710}{487621}a^{3}-\frac{2017134}{487621}a^{2}+\frac{1427642}{487621}a-\frac{306901}{487621}$, $\frac{250041}{487621}a^{15}-\frac{1144644}{487621}a^{14}+\frac{3161777}{487621}a^{13}-\frac{6619066}{487621}a^{12}+\frac{12549256}{487621}a^{11}-\frac{21744292}{487621}a^{10}+\frac{33756915}{487621}a^{9}-\frac{45582877}{487621}a^{8}+\frac{51922441}{487621}a^{7}-\frac{48344280}{487621}a^{6}+\frac{35222750}{487621}a^{5}-\frac{19802506}{487621}a^{4}+\frac{9033354}{487621}a^{3}-\frac{4715525}{487621}a^{2}+\frac{2377537}{487621}a-\frac{790161}{487621}$, $\frac{322766}{487621}a^{15}-\frac{1552517}{487621}a^{14}+\frac{3831183}{487621}a^{13}-\frac{7422038}{487621}a^{12}+\frac{13601081}{487621}a^{11}-\frac{23208556}{487621}a^{10}+\frac{34424711}{487621}a^{9}-\frac{43081561}{487621}a^{8}+\frac{44352522}{487621}a^{7}-\frac{35002443}{487621}a^{6}+\frac{18986071}{487621}a^{5}-\frac{7000793}{487621}a^{4}+\frac{3573824}{487621}a^{3}-\frac{2594195}{487621}a^{2}+\frac{225817}{487621}a-\frac{32927}{487621}$, $\frac{505949}{487621}a^{15}-\frac{1729330}{487621}a^{14}+\frac{4120407}{487621}a^{13}-\frac{7975327}{487621}a^{12}+\frac{14706304}{487621}a^{11}-\frac{24428385}{487621}a^{10}+\frac{35430030}{487621}a^{9}-\frac{44018462}{487621}a^{8}+\frac{44730148}{487621}a^{7}-\frac{36088930}{487621}a^{6}+\frac{21409837}{487621}a^{5}-\frac{11231620}{487621}a^{4}+\frac{5637462}{487621}a^{3}-\frac{3834632}{487621}a^{2}+\frac{1493265}{487621}a-\frac{612838}{487621}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 112.874368042 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 112.874368042 \cdot 1}{30\cdot\sqrt{3243658447265625}}\cr\approx \mathstrut & 0.160470595877 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 8 sibling: 8.4.56953125.1
Minimal sibling: 8.4.56953125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ R R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.16.12.2$x^{16} + 12 x^{12} + 36 x^{8} + 324$$4$$4$$12$$C_8: C_2$$[\ ]_{4}^{4}$
\(5\) Copy content Toggle raw display 5.16.14.1$x^{16} - 20 x^{8} - 100$$8$$2$$14$$C_8: C_2$$[\ ]_{8}^{2}$