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Properties

Label 18.2.116...689.1
Degree $18$
Signature $[2, 8]$
Discriminant $1.168\times 10^{19}$
Root discriminant \(11.46\)
Ramified primes $5881,44555813$
Class number $1$
Class group trivial
Galois group $C_2^9.S_9$ (as 18T968)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 4*x^16 + 9*x^15 - 27*x^14 + 17*x^13 + 26*x^12 - 46*x^11 + 12*x^10 + 20*x^9 - 12*x^8 + 15*x^7 - 48*x^6 + 48*x^5 - x^4 - 29*x^3 + 20*x^2 - 5*x + 1)
 
gp: K = bnfinit(y^18 - 4*y^17 + 4*y^16 + 9*y^15 - 27*y^14 + 17*y^13 + 26*y^12 - 46*y^11 + 12*y^10 + 20*y^9 - 12*y^8 + 15*y^7 - 48*y^6 + 48*y^5 - y^4 - 29*y^3 + 20*y^2 - 5*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^17 + 4*x^16 + 9*x^15 - 27*x^14 + 17*x^13 + 26*x^12 - 46*x^11 + 12*x^10 + 20*x^9 - 12*x^8 + 15*x^7 - 48*x^6 + 48*x^5 - x^4 - 29*x^3 + 20*x^2 - 5*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 4*x^16 + 9*x^15 - 27*x^14 + 17*x^13 + 26*x^12 - 46*x^11 + 12*x^10 + 20*x^9 - 12*x^8 + 15*x^7 - 48*x^6 + 48*x^5 - x^4 - 29*x^3 + 20*x^2 - 5*x + 1)
 

\( x^{18} - 4 x^{17} + 4 x^{16} + 9 x^{15} - 27 x^{14} + 17 x^{13} + 26 x^{12} - 46 x^{11} + 12 x^{10} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(11675081596366988689\) \(\medspace = 5881\cdot 44555813^{2}\) 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:  \(11.46\)
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:  $5881^{1/2}44555813^{1/2}\approx 511891.332465202$
Ramified primes:   \(5881\), \(44555813\) 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:  \(\Q(\sqrt{5881}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7}a^{16}+\frac{2}{7}a^{14}+\frac{3}{7}a^{13}+\frac{2}{7}a^{12}-\frac{2}{7}a^{11}-\frac{2}{7}a^{8}-\frac{2}{7}a^{7}-\frac{2}{7}a^{6}-\frac{3}{7}a^{5}-\frac{2}{7}a^{3}-\frac{2}{7}a^{2}+\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{919667}a^{17}+\frac{29147}{919667}a^{16}+\frac{286036}{919667}a^{15}-\frac{385244}{919667}a^{14}-\frac{325515}{919667}a^{13}+\frac{299120}{919667}a^{12}+\frac{152938}{919667}a^{11}-\frac{16949}{131381}a^{10}+\frac{305506}{919667}a^{9}-\frac{35686}{131381}a^{8}+\frac{10939}{131381}a^{7}+\frac{279110}{919667}a^{6}+\frac{172994}{919667}a^{5}+\frac{151219}{919667}a^{4}+\frac{31591}{131381}a^{3}-\frac{369631}{919667}a^{2}-\frac{163308}{919667}a+\frac{393165}{919667}$ 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:  $9$
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:   $\frac{84718}{131381}a^{17}-\frac{1394872}{919667}a^{16}-\frac{39316}{131381}a^{15}+\frac{5281606}{919667}a^{14}-\frac{6141711}{919667}a^{13}-\frac{2718365}{919667}a^{12}+\frac{10212376}{919667}a^{11}-\frac{682555}{131381}a^{10}-\frac{331073}{131381}a^{9}+\frac{1405032}{919667}a^{8}-\frac{2851038}{919667}a^{7}+\frac{10301395}{919667}a^{6}-\frac{11001058}{919667}a^{5}+\frac{404075}{131381}a^{4}+\frac{3359222}{919667}a^{3}-\frac{4995768}{919667}a^{2}+\frac{3716238}{919667}a-\frac{720599}{919667}$, $\frac{62447}{131381}a^{17}-\frac{1250084}{919667}a^{16}+\frac{54856}{131381}a^{15}+\frac{3892372}{919667}a^{14}-\frac{6554816}{919667}a^{13}+\frac{764898}{919667}a^{12}+\frac{8164631}{919667}a^{11}-\frac{1113117}{131381}a^{10}+\frac{98172}{131381}a^{9}+\frac{2407978}{919667}a^{8}-\frac{3074359}{919667}a^{7}+\frac{7335495}{919667}a^{6}-\frac{11011901}{919667}a^{5}+\frac{1214566}{131381}a^{4}-\frac{277173}{919667}a^{3}-\frac{6046314}{919667}a^{2}+\frac{4721620}{919667}a-\frac{918055}{919667}$, $\frac{141975}{131381}a^{17}-\frac{3281511}{919667}a^{16}+\frac{225381}{131381}a^{15}+\frac{9990235}{919667}a^{14}-\frac{19040818}{919667}a^{13}+\frac{2794683}{919667}a^{12}+\frac{25730674}{919667}a^{11}-\frac{3397821}{131381}a^{10}-\frac{565895}{131381}a^{9}+\frac{10808907}{919667}a^{8}-\frac{2545287}{919667}a^{7}+\frac{17424789}{919667}a^{6}-\frac{35807087}{919667}a^{5}+\frac{2844554}{131381}a^{4}+\frac{8509610}{919667}a^{3}-\frac{16317864}{919667}a^{2}+\frac{9039824}{919667}a-\frac{2106308}{919667}$, $\frac{532023}{919667}a^{17}-\frac{228803}{131381}a^{16}+\frac{432338}{919667}a^{15}+\frac{5413582}{919667}a^{14}-\frac{8564888}{919667}a^{13}-\frac{720682}{919667}a^{12}+\frac{1864788}{131381}a^{11}-\frac{1236702}{131381}a^{10}-\frac{3887608}{919667}a^{9}+\frac{3690487}{919667}a^{8}-\frac{548825}{919667}a^{7}+\frac{10206141}{919667}a^{6}-\frac{2281099}{131381}a^{5}+\frac{5954546}{919667}a^{4}+\frac{5510606}{919667}a^{3}-\frac{5973810}{919667}a^{2}+\frac{3504313}{919667}a-\frac{63411}{131381}$, $\frac{70081}{131381}a^{17}-\frac{1477215}{919667}a^{16}+\frac{101460}{131381}a^{15}+\frac{4232057}{919667}a^{14}-\frac{8288939}{919667}a^{13}+\frac{2509427}{919667}a^{12}+\frac{9361418}{919667}a^{11}-\frac{1618689}{131381}a^{10}+\frac{318226}{131381}a^{9}+\frac{4031779}{919667}a^{8}-\frac{3947297}{919667}a^{7}+\frac{7913174}{919667}a^{6}-\frac{14204357}{919667}a^{5}+\frac{1701089}{131381}a^{4}-\frac{222212}{919667}a^{3}-\frac{6962295}{919667}a^{2}+\frac{5771308}{919667}a-\frac{1153543}{919667}$, $\frac{215991}{919667}a^{17}-\frac{813367}{919667}a^{16}+\frac{731617}{919667}a^{15}+\frac{243976}{131381}a^{14}-\frac{5154836}{919667}a^{13}+\frac{3774314}{919667}a^{12}+\frac{3916777}{919667}a^{11}-\frac{1345085}{131381}a^{10}+\frac{5037531}{919667}a^{9}+\frac{3324299}{919667}a^{8}-\frac{5204963}{919667}a^{7}+\frac{491574}{131381}a^{6}-\frac{7492206}{919667}a^{5}+\frac{11005528}{919667}a^{4}-\frac{3376689}{919667}a^{3}-\frac{6589196}{919667}a^{2}+\frac{6681702}{919667}a-\frac{1160979}{919667}$, $\frac{305429}{919667}a^{17}-\frac{1088545}{919667}a^{16}+\frac{842446}{919667}a^{15}+\frac{2761544}{919667}a^{14}-\frac{7032045}{919667}a^{13}+\frac{3618606}{919667}a^{12}+\frac{7594236}{919667}a^{11}-\frac{1749912}{131381}a^{10}+\frac{2817588}{919667}a^{9}+\frac{6419360}{919667}a^{8}-\frac{4772233}{919667}a^{7}+\frac{3697055}{919667}a^{6}-\frac{11805253}{919667}a^{5}+\frac{12027215}{919667}a^{4}+\frac{1571055}{919667}a^{3}-\frac{10318355}{919667}a^{2}+\frac{759360}{131381}a-\frac{211930}{919667}$, $\frac{787331}{919667}a^{17}-\frac{1690566}{919667}a^{16}-\frac{1205710}{919667}a^{15}+\frac{7792699}{919667}a^{14}-\frac{5498623}{919667}a^{13}-\frac{9628119}{919667}a^{12}+\frac{14097649}{919667}a^{11}+\frac{551956}{131381}a^{10}-\frac{10077366}{919667}a^{9}-\frac{4756702}{919667}a^{8}+\frac{1759955}{919667}a^{7}+\frac{18152577}{919667}a^{6}-\frac{10967976}{919667}a^{5}-\frac{8960334}{919667}a^{4}+\frac{7169387}{919667}a^{3}+\frac{1553430}{919667}a^{2}+\frac{1717846}{919667}a-\frac{511201}{919667}$, $\frac{284380}{919667}a^{17}-\frac{1317240}{919667}a^{16}+\frac{1130531}{919667}a^{15}+\frac{3715466}{919667}a^{14}-\frac{1288491}{131381}a^{13}+\frac{2299579}{919667}a^{12}+\frac{13017538}{919667}a^{11}-\frac{1821207}{131381}a^{10}-\frac{4823878}{919667}a^{9}+\frac{8867515}{919667}a^{8}+\frac{1399705}{919667}a^{7}+\frac{3806223}{919667}a^{6}-\frac{17318165}{919667}a^{5}+\frac{9226970}{919667}a^{4}+\frac{7993460}{919667}a^{3}-\frac{9155827}{919667}a^{2}+\frac{1128936}{919667}a+\frac{535461}{919667}$ 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:  \( 111.7260845 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 111.7260845 \cdot 1}{2\cdot\sqrt{11675081596366988689}}\cr\approx \mathstrut & 0.1588523190 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 4*x^16 + 9*x^15 - 27*x^14 + 17*x^13 + 26*x^12 - 46*x^11 + 12*x^10 + 20*x^9 - 12*x^8 + 15*x^7 - 48*x^6 + 48*x^5 - x^4 - 29*x^3 + 20*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^18 - 4*x^17 + 4*x^16 + 9*x^15 - 27*x^14 + 17*x^13 + 26*x^12 - 46*x^11 + 12*x^10 + 20*x^9 - 12*x^8 + 15*x^7 - 48*x^6 + 48*x^5 - x^4 - 29*x^3 + 20*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^18 - 4*x^17 + 4*x^16 + 9*x^15 - 27*x^14 + 17*x^13 + 26*x^12 - 46*x^11 + 12*x^10 + 20*x^9 - 12*x^8 + 15*x^7 - 48*x^6 + 48*x^5 - x^4 - 29*x^3 + 20*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^18 - 4*x^17 + 4*x^16 + 9*x^15 - 27*x^14 + 17*x^13 + 26*x^12 - 46*x^11 + 12*x^10 + 20*x^9 - 12*x^8 + 15*x^7 - 48*x^6 + 48*x^5 - x^4 - 29*x^3 + 20*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

$C_2^9.S_9$ (as 18T968):

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 185794560
The 300 conjugacy class representatives for $C_2^9.S_9$
Character table for $C_2^9.S_9$

Intermediate fields

9.1.44555813.1

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 18 sibling: data not computed
Degree 36 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 ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.8.0.1}{8} }$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.8.0.1}{8} }$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ $18$ ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $18$ $18$

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
\(5881\) Copy content Toggle raw display $\Q_{5881}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{5881}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(44555813\) Copy content Toggle raw display $\Q_{44555813}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{44555813}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$2$$2$$2$