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Properties

Label 16.0.38416000000000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.842\times 10^{16}$
Root discriminant \(10.88\)
Ramified primes $2,5,7$
Class number $1$
Class group trivial
Galois group $C_4\times S_4$ (as 16T181)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 164*x^12 - 256*x^11 + 344*x^10 - 422*x^9 + 478*x^8 - 480*x^7 + 400*x^6 - 260*x^5 + 127*x^4 - 46*x^3 + 14*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^16 - 8*y^15 + 32*y^14 - 84*y^13 + 164*y^12 - 256*y^11 + 344*y^10 - 422*y^9 + 478*y^8 - 480*y^7 + 400*y^6 - 260*y^5 + 127*y^4 - 46*y^3 + 14*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 164*x^12 - 256*x^11 + 344*x^10 - 422*x^9 + 478*x^8 - 480*x^7 + 400*x^6 - 260*x^5 + 127*x^4 - 46*x^3 + 14*x^2 - 4*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 164*x^12 - 256*x^11 + 344*x^10 - 422*x^9 + 478*x^8 - 480*x^7 + 400*x^6 - 260*x^5 + 127*x^4 - 46*x^3 + 14*x^2 - 4*x + 1)
 

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 164 x^{12} - 256 x^{11} + 344 x^{10} - 422 x^{9} + 478 x^{8} + \cdots + 1 \) 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:   \(38416000000000000\) \(\medspace = 2^{16}\cdot 5^{12}\cdot 7^{4}\) 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:  \(10.88\)
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:  $2^{7/6}5^{3/4}7^{1/2}\approx 19.85995155932405$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11}a^{14}+\frac{4}{11}a^{13}+\frac{4}{11}a^{12}+\frac{1}{11}a^{11}+\frac{4}{11}a^{10}+\frac{2}{11}a^{9}-\frac{2}{11}a^{8}-\frac{4}{11}a^{7}-\frac{1}{11}a^{6}-\frac{1}{11}a^{5}+\frac{2}{11}a^{4}+\frac{5}{11}a^{3}+\frac{2}{11}a^{2}+\frac{5}{11}a-\frac{1}{11}$, $\frac{1}{1661}a^{15}+\frac{68}{1661}a^{14}+\frac{821}{1661}a^{13}-\frac{51}{1661}a^{12}+\frac{365}{1661}a^{11}-\frac{149}{1661}a^{10}-\frac{259}{1661}a^{9}-\frac{57}{151}a^{8}-\frac{213}{1661}a^{7}-\frac{813}{1661}a^{6}-\frac{535}{1661}a^{5}+\frac{1}{1661}a^{4}-\frac{250}{1661}a^{3}+\frac{584}{1661}a^{2}+\frac{69}{151}a-\frac{757}{1661}$ 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:   \( \frac{1000}{1661} a^{15} - \frac{4480}{1661} a^{14} + \frac{564}{151} a^{13} + \frac{9551}{1661} a^{12} - \frac{56290}{1661} a^{11} + \frac{129142}{1661} a^{10} - \frac{197996}{1661} a^{9} + \frac{255444}{1661} a^{8} - \frac{321720}{1661} a^{7} + \frac{34756}{151} a^{6} - \frac{379470}{1661} a^{5} + \frac{274612}{1661} a^{4} - \frac{127388}{1661} a^{3} + \frac{40400}{1661} a^{2} - \frac{12006}{1661} a + \frac{6456}{1661} \)  (order $10$) 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:   $\frac{1081}{1661}a^{15}-\frac{7730}{1661}a^{14}+\frac{27707}{1661}a^{13}-\frac{5863}{151}a^{12}+\frac{110685}{1661}a^{11}-\frac{150499}{1661}a^{10}+\frac{180420}{1661}a^{9}-\frac{204704}{1661}a^{8}+\frac{19481}{151}a^{7}-\frac{183045}{1661}a^{6}+\frac{104185}{1661}a^{5}-\frac{21871}{1661}a^{4}-\frac{15362}{1661}a^{3}+\frac{12053}{1661}a^{2}-\frac{961}{1661}a+\frac{405}{1661}$, $\frac{3336}{1661}a^{15}-\frac{26228}{1661}a^{14}+\frac{102094}{1661}a^{13}-\frac{258924}{1661}a^{12}+\frac{486196}{1661}a^{11}-\frac{728698}{1661}a^{10}+\frac{945258}{1661}a^{9}-\frac{1133728}{1661}a^{8}+\frac{1258472}{1661}a^{7}-\frac{1216664}{1661}a^{6}+\frac{85595}{151}a^{5}-\frac{532714}{1661}a^{4}+\frac{209410}{1661}a^{3}-\frac{59472}{1661}a^{2}+\frac{17572}{1661}a-\frac{6672}{1661}$, $\frac{2660}{1661}a^{15}-\frac{2198}{151}a^{14}+\frac{104590}{1661}a^{13}-\frac{288170}{1661}a^{12}+\frac{52257}{151}a^{11}-\frac{900982}{1661}a^{10}+\frac{1196446}{1661}a^{9}-\frac{1453702}{1661}a^{8}+\frac{1648892}{1661}a^{7}-\frac{1656881}{1661}a^{6}+\frac{1354847}{1661}a^{5}-\frac{824367}{1661}a^{4}+\frac{346096}{1661}a^{3}-\frac{102426}{1661}a^{2}+\frac{26948}{1661}a-\frac{9699}{1661}$, $\frac{6856}{1661}a^{15}-\frac{52024}{1661}a^{14}+\frac{195645}{1661}a^{13}-\frac{479214}{1661}a^{12}+\frac{871338}{1661}a^{11}-\frac{1270694}{1661}a^{10}+\frac{1622702}{1661}a^{9}-\frac{175768}{151}a^{8}+\frac{2120788}{1661}a^{7}-\frac{1996134}{1661}a^{6}+\frac{1476157}{1661}a^{5}-\frac{787102}{1661}a^{4}+\frac{295810}{1661}a^{3}-\frac{85478}{1661}a^{2}+\frac{2548}{151}a-\frac{7672}{1661}$, $\frac{5877}{1661}a^{15}-\frac{40680}{1661}a^{14}+\frac{12763}{151}a^{13}-\frac{315280}{1661}a^{12}+\frac{528801}{1661}a^{11}-\frac{716821}{1661}a^{10}+\frac{877700}{1661}a^{9}-\frac{1021994}{1661}a^{8}+\frac{1074203}{1661}a^{7}-\frac{83728}{151}a^{6}+\frac{569952}{1661}a^{5}-\frac{221982}{1661}a^{4}+\frac{56454}{1661}a^{3}-\frac{14709}{1661}a^{2}+\frac{10069}{1661}a+\frac{1081}{1661}$, $\frac{7937}{1661}a^{15}-\frac{59754}{1661}a^{14}+\frac{223352}{1661}a^{13}-\frac{543707}{1661}a^{12}+\frac{982023}{1661}a^{11}-\frac{1421193}{1661}a^{10}+\frac{1803122}{1661}a^{9}-\frac{2138152}{1661}a^{8}+\frac{2335079}{1661}a^{7}-\frac{2179179}{1661}a^{6}+\frac{1580342}{1661}a^{5}-\frac{73543}{151}a^{4}+\frac{280448}{1661}a^{3}-\frac{6675}{151}a^{2}+\frac{28728}{1661}a-\frac{8928}{1661}$, $\frac{8315}{1661}a^{15}-\frac{62438}{1661}a^{14}+\frac{232445}{1661}a^{13}-\frac{563589}{1661}a^{12}+\frac{1015199}{1661}a^{11}-\frac{1468153}{1661}a^{10}+\frac{1864374}{1661}a^{9}-\frac{201249}{151}a^{8}+\frac{2416286}{1661}a^{7}-\frac{2252141}{1661}a^{6}+\frac{1635718}{1661}a^{5}-\frac{845439}{1661}a^{4}+\frac{301463}{1661}a^{3}-\frac{82193}{1661}a^{2}+\frac{2502}{151}a-\frac{7570}{1661}$ 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:  \( 175.447877376 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 175.447877376 \cdot 1}{10\cdot\sqrt{38416000000000000}}\cr\approx \mathstrut & 0.217435771583 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 164*x^12 - 256*x^11 + 344*x^10 - 422*x^9 + 478*x^8 - 480*x^7 + 400*x^6 - 260*x^5 + 127*x^4 - 46*x^3 + 14*x^2 - 4*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 - 8*x^15 + 32*x^14 - 84*x^13 + 164*x^12 - 256*x^11 + 344*x^10 - 422*x^9 + 478*x^8 - 480*x^7 + 400*x^6 - 260*x^5 + 127*x^4 - 46*x^3 + 14*x^2 - 4*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 - 8*x^15 + 32*x^14 - 84*x^13 + 164*x^12 - 256*x^11 + 344*x^10 - 422*x^9 + 478*x^8 - 480*x^7 + 400*x^6 - 260*x^5 + 127*x^4 - 46*x^3 + 14*x^2 - 4*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 - 8*x^15 + 32*x^14 - 84*x^13 + 164*x^12 - 256*x^11 + 344*x^10 - 422*x^9 + 478*x^8 - 480*x^7 + 400*x^6 - 260*x^5 + 127*x^4 - 46*x^3 + 14*x^2 - 4*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

$C_4\times S_4$ (as 16T181):

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 96
The 20 conjugacy class representatives for $C_4\times S_4$
Character table for $C_4\times S_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.14000.1, 8.4.196000000.1

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 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Minimal sibling: 12.0.768320000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ R R ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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

# 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
\(2\) Copy content Toggle raw display Deg $16$$4$$4$$16$
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(7\) Copy content Toggle raw display 7.4.0.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$