↑

Properties

Label 16.0.5190614520439741.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.191\times 10^{15}$
Root discriminant \(9.60\)
Ramified primes $617,2389$
Class number $1$
Class group trivial
Galois group $C_2^8.S_8$ (as 16T1948)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 15*x^12 - 15*x^11 + 12*x^10 - 5*x^9 - 2*x^8 + 4*x^7 - 5*x^6 + 6*x^5 + 3*x^4 - 5*x^3 - x^2 + 1)
 
gp: K = bnfinit(y^16 - 3*y^15 + 7*y^14 - 12*y^13 + 15*y^12 - 15*y^11 + 12*y^10 - 5*y^9 - 2*y^8 + 4*y^7 - 5*y^6 + 6*y^5 + 3*y^4 - 5*y^3 - y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 15*x^12 - 15*x^11 + 12*x^10 - 5*x^9 - 2*x^8 + 4*x^7 - 5*x^6 + 6*x^5 + 3*x^4 - 5*x^3 - x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 15*x^12 - 15*x^11 + 12*x^10 - 5*x^9 - 2*x^8 + 4*x^7 - 5*x^6 + 6*x^5 + 3*x^4 - 5*x^3 - x^2 + 1)
 

\( x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 15 x^{12} - 15 x^{11} + 12 x^{10} - 5 x^{9} - 2 x^{8} + \cdots + 1 \) 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:   \(5190614520439741\) \(\medspace = 617^{2}\cdot 2389^{3}\) 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:  \(9.60\)
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:  $617^{1/2}2389^{3/4}\approx 8487.986777006932$
Ramified primes:   \(617\), \(2389\) 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:  \(\Q(\sqrt{2389}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{692467}a^{15}-\frac{330365}{692467}a^{14}+\frac{318267}{692467}a^{13}+\frac{174147}{692467}a^{12}-\frac{7905}{692467}a^{11}+\frac{218538}{692467}a^{10}-\frac{41324}{692467}a^{9}-\frac{107622}{692467}a^{8}+\frac{193514}{692467}a^{7}+\frac{266310}{692467}a^{6}-\frac{79408}{692467}a^{5}-\frac{34126}{692467}a^{4}-\frac{121612}{692467}a^{3}-\frac{259334}{692467}a^{2}+\frac{4266}{692467}a-\frac{153947}{692467}$ 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:   \( -1 \)  (order $2$) 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:   $\frac{176077}{692467}a^{15}-\frac{372704}{692467}a^{14}+\frac{914117}{692467}a^{13}-\frac{1234842}{692467}a^{12}+\frac{1354919}{692467}a^{11}-\frac{875764}{692467}a^{10}+\frac{237288}{692467}a^{9}+\frac{985495}{692467}a^{8}-\frac{1551558}{692467}a^{7}+\frac{1355432}{692467}a^{6}-\frac{1013686}{692467}a^{5}+\frac{424924}{692467}a^{4}+\frac{1465851}{692467}a^{3}-\frac{786271}{692467}a^{2}+\frac{510254}{692467}a+\frac{94796}{692467}$, $\frac{372282}{692467}a^{15}-\frac{571527}{692467}a^{14}+\frac{1201726}{692467}a^{13}-\frac{1121888}{692467}a^{12}+\frac{95540}{692467}a^{11}+\frac{1200820}{692467}a^{10}-\frac{2411897}{692467}a^{9}+\frac{3869551}{692467}a^{8}-\frac{3180199}{692467}a^{7}+\frac{1226563}{692467}a^{6}-\frac{752826}{692467}a^{5}+\frac{196517}{692467}a^{4}+\frac{3688678}{692467}a^{3}+\frac{446353}{692467}a^{2}-\frac{1749220}{692467}a-\frac{358266}{692467}$, $\frac{103175}{692467}a^{15}-\frac{105734}{692467}a^{14}+\frac{412585}{692467}a^{13}-\frac{516991}{692467}a^{12}+\frac{820218}{692467}a^{11}-\frac{1144771}{692467}a^{10}+\frac{1300553}{692467}a^{9}-\frac{883972}{692467}a^{8}+\frac{598406}{692467}a^{7}+\frac{136157}{692467}a^{6}-\frac{1035790}{692467}a^{5}+\frac{937112}{692467}a^{4}+\frac{183940}{692467}a^{3}+\frac{1524364}{692467}a^{2}+\frac{428005}{692467}a-\frac{366146}{692467}$, $\frac{422988}{692467}a^{15}-\frac{590020}{692467}a^{14}+\frac{1304793}{692467}a^{13}-\frac{1163290}{692467}a^{12}+\frac{203003}{692467}a^{11}+\frac{839247}{692467}a^{10}-\frac{1689032}{692467}a^{9}+\frac{2735912}{692467}a^{8}-\frac{1731771}{692467}a^{7}-\frac{442478}{692467}a^{6}+\frac{173198}{692467}a^{5}-\frac{413873}{692467}a^{4}+\frac{4341708}{692467}a^{3}+\frac{1297346}{692467}a^{2}-\frac{1487128}{692467}a-\frac{906824}{692467}$, $\frac{525964}{692467}a^{15}-\frac{1429951}{692467}a^{14}+\frac{3474143}{692467}a^{13}-\frac{5866986}{692467}a^{12}+\frac{7443585}{692467}a^{11}-\frac{7786302}{692467}a^{10}+\frac{6450063}{692467}a^{9}-\frac{3045028}{692467}a^{8}-\frac{172032}{692467}a^{7}+\frac{1402882}{692467}a^{6}-\frac{2372075}{692467}a^{5}+\frac{2467044}{692467}a^{4}+\frac{2412690}{692467}a^{3}-\frac{1660651}{692467}a^{2}+\frac{169344}{692467}a-\frac{413598}{692467}$, $\frac{53145}{692467}a^{15}+\frac{252860}{692467}a^{14}-\frac{591694}{692467}a^{13}+\frac{1605794}{692467}a^{12}-\frac{2553624}{692467}a^{11}+\frac{2915354}{692467}a^{10}-\frac{2428524}{692467}a^{9}+\frac{1591164}{692467}a^{8}+\frac{474113}{692467}a^{7}-\frac{1672997}{692467}a^{6}+\frac{1140672}{692467}a^{5}-\frac{747664}{692467}a^{4}+\frac{1109705}{692467}a^{3}+\frac{2635139}{692467}a^{2}-\frac{1105073}{692467}a-\frac{708177}{692467}$, $\frac{557797}{692467}a^{15}-\frac{1442667}{692467}a^{14}+\frac{3382877}{692467}a^{13}-\frac{5443870}{692467}a^{12}+\frac{6476774}{692467}a^{11}-\frac{6204696}{692467}a^{10}+\frac{4593070}{692467}a^{9}-\frac{1264504}{692467}a^{8}-\frac{1612236}{692467}a^{7}+\frac{1668098}{692467}a^{6}-\frac{1969922}{692467}a^{5}+\frac{1922342}{692467}a^{4}+\frac{3413358}{692467}a^{3}-\frac{2140766}{692467}a^{2}-\frac{447077}{692467}a-\frac{419490}{692467}$ 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:  \( 10.0160978095 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10.0160978095 \cdot 1}{2\cdot\sqrt{5190614520439741}}\cr\approx \mathstrut & 0.168848854797 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 15*x^12 - 15*x^11 + 12*x^10 - 5*x^9 - 2*x^8 + 4*x^7 - 5*x^6 + 6*x^5 + 3*x^4 - 5*x^3 - x^2 + 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 + 7*x^14 - 12*x^13 + 15*x^12 - 15*x^11 + 12*x^10 - 5*x^9 - 2*x^8 + 4*x^7 - 5*x^6 + 6*x^5 + 3*x^4 - 5*x^3 - x^2 + 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 + 7*x^14 - 12*x^13 + 15*x^12 - 15*x^11 + 12*x^10 - 5*x^9 - 2*x^8 + 4*x^7 - 5*x^6 + 6*x^5 + 3*x^4 - 5*x^3 - x^2 + 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 + 7*x^14 - 12*x^13 + 15*x^12 - 15*x^11 + 12*x^10 - 5*x^9 - 2*x^8 + 4*x^7 - 5*x^6 + 6*x^5 + 3*x^4 - 5*x^3 - x^2 + 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

$C_2^8.S_8$ (as 16T1948):

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 non-solvable group of order 10321920
The 185 conjugacy class representatives for $C_2^8.S_8$
Character table for $C_2^8.S_8$

Intermediate fields

8.0.1474013.1

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 $16$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.7.0.1}{7} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ $16$ ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.7.0.1}{7} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(617\) Copy content Toggle raw display $\Q_{617}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{617}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $4$$2$$2$$2$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
\(2389\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$4$$1$$3$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$