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Properties

Label 16.0.4393378612917909.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.393\times 10^{15}$
Root discriminant \(9.50\)
Ramified primes $3,61,104773$
Class number $1$
Class group trivial
Galois group $C_2^6.S_4^2:D_4$ (as 16T1905)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 5*x^14 - 6*x^13 + 6*x^12 - 6*x^11 + 4*x^10 - x^8 + 4*x^6 + 6*x^5 + 6*x^4 + 6*x^3 + 5*x^2 + 3*x + 1)
 
gp: K = bnfinit(y^16 - 3*y^15 + 5*y^14 - 6*y^13 + 6*y^12 - 6*y^11 + 4*y^10 - y^8 + 4*y^6 + 6*y^5 + 6*y^4 + 6*y^3 + 5*y^2 + 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 5*x^14 - 6*x^13 + 6*x^12 - 6*x^11 + 4*x^10 - x^8 + 4*x^6 + 6*x^5 + 6*x^4 + 6*x^3 + 5*x^2 + 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 5*x^14 - 6*x^13 + 6*x^12 - 6*x^11 + 4*x^10 - x^8 + 4*x^6 + 6*x^5 + 6*x^4 + 6*x^3 + 5*x^2 + 3*x + 1)
 

\( x^{16} - 3 x^{15} + 5 x^{14} - 6 x^{13} + 6 x^{12} - 6 x^{11} + 4 x^{10} - x^{8} + 4 x^{6} + 6 x^{5} + \cdots + 1 \) 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:   \(4393378612917909\) \(\medspace = 3^{8}\cdot 61\cdot 104773^{2}\) 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:  \(9.50\)
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:  $3^{1/2}61^{1/2}104773^{1/2}\approx 4378.75084927197$
Ramified primes:   \(3\), \(61\), \(104773\) 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:  \(\Q(\sqrt{61}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{779}a^{14}-\frac{172}{779}a^{13}+\frac{251}{779}a^{12}+\frac{248}{779}a^{11}-\frac{368}{779}a^{10}+\frac{6}{41}a^{9}-\frac{155}{779}a^{8}-\frac{177}{779}a^{7}+\frac{155}{779}a^{6}+\frac{6}{41}a^{5}+\frac{368}{779}a^{4}+\frac{248}{779}a^{3}-\frac{251}{779}a^{2}-\frac{172}{779}a-\frac{1}{779}$, $\frac{1}{779}a^{15}+\frac{269}{779}a^{13}-\frac{204}{779}a^{12}+\frac{222}{779}a^{11}-\frac{83}{779}a^{10}-\frac{22}{779}a^{9}-\frac{351}{779}a^{8}+\frac{92}{779}a^{7}+\frac{288}{779}a^{6}-\frac{278}{779}a^{5}-\frac{334}{779}a^{4}+\frac{339}{779}a^{3}+\frac{280}{779}a^{2}+\frac{17}{779}a-\frac{172}{779}$ 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:   \( \frac{9}{41} a^{14} - \frac{31}{41} a^{13} + \frac{45}{41} a^{12} - \frac{23}{41} a^{11} - \frac{32}{41} a^{10} + \frac{83}{41} a^{9} - \frac{124}{41} a^{8} + \frac{170}{41} a^{7} - \frac{163}{41} a^{6} + \frac{83}{41} a^{5} + \frac{32}{41} a^{4} + \frac{18}{41} a^{3} - \frac{4}{41} a^{2} + \frac{10}{41} a + \frac{32}{41} \)  (order $6$) 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:   $\frac{9}{41}a^{15}-\frac{31}{41}a^{14}+\frac{45}{41}a^{13}-\frac{23}{41}a^{12}-\frac{32}{41}a^{11}+\frac{83}{41}a^{10}-\frac{124}{41}a^{9}+\frac{170}{41}a^{8}-\frac{163}{41}a^{7}+\frac{83}{41}a^{6}+\frac{32}{41}a^{5}+\frac{18}{41}a^{4}-\frac{4}{41}a^{3}+\frac{10}{41}a^{2}-\frac{9}{41}a$, $\frac{320}{779}a^{15}-\frac{967}{779}a^{14}+\frac{1566}{779}a^{13}-\frac{1850}{779}a^{12}+\frac{1825}{779}a^{11}-\frac{1779}{779}a^{10}+\frac{1130}{779}a^{9}+\frac{173}{779}a^{8}-\frac{383}{779}a^{7}-\frac{79}{779}a^{6}+\frac{1005}{779}a^{5}+\frac{2327}{779}a^{4}+\frac{1094}{779}a^{3}+\frac{2021}{779}a^{2}+\frac{384}{779}a+\frac{457}{779}$, $\frac{127}{779}a^{15}-\frac{305}{779}a^{14}+\frac{154}{779}a^{13}+\frac{365}{779}a^{12}-\frac{706}{779}a^{11}+\frac{429}{779}a^{10}-\frac{172}{779}a^{9}+\frac{19}{41}a^{8}+\frac{233}{779}a^{7}-\frac{1351}{779}a^{6}+\frac{1592}{779}a^{5}+\frac{1142}{779}a^{4}+\frac{131}{779}a^{3}-\frac{840}{779}a^{2}+\frac{868}{779}a+\frac{273}{779}$, $\frac{411}{779}a^{15}-\frac{1359}{779}a^{14}+\frac{2326}{779}a^{13}-\frac{2735}{779}a^{12}+\frac{2711}{779}a^{11}-\frac{2959}{779}a^{10}+\frac{2738}{779}a^{9}-\frac{1389}{779}a^{8}+\frac{1031}{779}a^{7}-\frac{1913}{779}a^{6}+\frac{3466}{779}a^{5}+\frac{34}{19}a^{4}+\frac{942}{779}a^{3}+\frac{2032}{779}a^{2}+\frac{803}{779}a+\frac{777}{779}$, $\frac{419}{779}a^{15}-\frac{1049}{779}a^{14}+\frac{1014}{779}a^{13}+\frac{217}{779}a^{12}-\frac{1986}{779}a^{11}+\frac{3042}{779}a^{10}-\frac{4164}{779}a^{9}+\frac{5399}{779}a^{8}-\frac{4026}{779}a^{7}+\frac{922}{779}a^{6}+\frac{2306}{779}a^{5}+\frac{2963}{779}a^{4}+\frac{1855}{779}a^{3}+\frac{2025}{779}a^{2}+\frac{2149}{779}a+\frac{1428}{779}$, $\frac{197}{779}a^{15}-\frac{381}{779}a^{14}+\frac{117}{779}a^{13}+\frac{506}{779}a^{12}-\frac{898}{779}a^{11}+\frac{775}{779}a^{10}-\frac{1028}{779}a^{9}+\frac{1593}{779}a^{8}-\frac{908}{779}a^{7}+\frac{18}{779}a^{6}-\frac{46}{779}a^{5}+\frac{3545}{779}a^{4}+\frac{339}{779}a^{3}+\frac{1223}{779}a^{2}+\frac{1108}{779}a+\frac{773}{779}$, $\frac{335}{779}a^{15}-\frac{878}{779}a^{14}+\frac{1199}{779}a^{13}-\frac{1267}{779}a^{12}+\frac{80}{41}a^{11}-\frac{2279}{779}a^{10}+\frac{2377}{779}a^{9}-\frac{1749}{779}a^{8}+\frac{2382}{779}a^{7}-\frac{2997}{779}a^{6}+\frac{3086}{779}a^{5}+\frac{2025}{779}a^{4}+\frac{2544}{779}a^{3}+\frac{1799}{779}a^{2}+\frac{1690}{779}a+\frac{904}{779}$ 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:  \( 27.0045913409 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 27.0045913409 \cdot 1}{6\cdot\sqrt{4393378612917909}}\cr\approx \mathstrut & 0.164939999998 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 5*x^14 - 6*x^13 + 6*x^12 - 6*x^11 + 4*x^10 - x^8 + 4*x^6 + 6*x^5 + 6*x^4 + 6*x^3 + 5*x^2 + 3*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 - 3*x^15 + 5*x^14 - 6*x^13 + 6*x^12 - 6*x^11 + 4*x^10 - x^8 + 4*x^6 + 6*x^5 + 6*x^4 + 6*x^3 + 5*x^2 + 3*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 - 3*x^15 + 5*x^14 - 6*x^13 + 6*x^12 - 6*x^11 + 4*x^10 - x^8 + 4*x^6 + 6*x^5 + 6*x^4 + 6*x^3 + 5*x^2 + 3*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 - 3*x^15 + 5*x^14 - 6*x^13 + 6*x^12 - 6*x^11 + 4*x^10 - x^8 + 4*x^6 + 6*x^5 + 6*x^4 + 6*x^3 + 5*x^2 + 3*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

$C_2^6.S_4^2:D_4$ (as 16T1905):

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 294912
The 230 conjugacy class representatives for $C_2^6.S_4^2:D_4$
Character table for $C_2^6.S_4^2:D_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), 8.0.8486613.1

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $16$ R ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ $16$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }$

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
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(61\) Copy content Toggle raw display $\Q_{61}$$x + 59$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 59$$1$$1$$0$Trivial$[\ ]$
61.2.1.1$x^{2} + 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
\(104773\) Copy content Toggle raw display $\Q_{104773}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{104773}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$