Properties

Label 18.0.167...571.1
Degree $18$
Signature $[0, 9]$
Discriminant $-1.678\times 10^{18}$
Root discriminant \(10.29\)
Ramified primes $139,367,299401$
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 - 2*x^17 + x^16 + 5*x^15 - 10*x^14 + 5*x^13 + 13*x^12 - 29*x^11 + 24*x^10 + 9*x^9 - 48*x^8 + 56*x^7 - 16*x^6 - 44*x^5 + 74*x^4 - 60*x^3 + 29*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^18 - 2*y^17 + y^16 + 5*y^15 - 10*y^14 + 5*y^13 + 13*y^12 - 29*y^11 + 24*y^10 + 9*y^9 - 48*y^8 + 56*y^7 - 16*y^6 - 44*y^5 + 74*y^4 - 60*y^3 + 29*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + x^16 + 5*x^15 - 10*x^14 + 5*x^13 + 13*x^12 - 29*x^11 + 24*x^10 + 9*x^9 - 48*x^8 + 56*x^7 - 16*x^6 - 44*x^5 + 74*x^4 - 60*x^3 + 29*x^2 - 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 + x^16 + 5*x^15 - 10*x^14 + 5*x^13 + 13*x^12 - 29*x^11 + 24*x^10 + 9*x^9 - 48*x^8 + 56*x^7 - 16*x^6 - 44*x^5 + 74*x^4 - 60*x^3 + 29*x^2 - 8*x + 1)
 

\( x^{18} - 2 x^{17} + x^{16} + 5 x^{15} - 10 x^{14} + 5 x^{13} + 13 x^{12} - 29 x^{11} + 24 x^{10} + 9 x^{9} - 48 x^{8} + 56 x^{7} - 16 x^{6} - 44 x^{5} + 74 x^{4} - 60 x^{3} + 29 x^{2} - 8 x + 1 \) 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:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1678237502892756571\) \(\medspace = -\,139\cdot 367^{2}\cdot 299401^{2}\) 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:  \(10.29\)
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:  $139^{1/2}367^{1/2}299401^{1/2}\approx 123585.36811856006$
Ramified primes:   \(139\), \(367\), \(299401\) 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:  \(\Q(\sqrt{-139}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{15073}a^{17}+\frac{1035}{15073}a^{16}+\frac{3113}{15073}a^{15}+\frac{2564}{15073}a^{14}+\frac{6010}{15073}a^{13}+\frac{7226}{15073}a^{12}+\frac{2094}{15073}a^{11}+\frac{937}{15073}a^{10}+\frac{7021}{15073}a^{9}+\frac{527}{15073}a^{8}+\frac{3823}{15073}a^{7}+\frac{308}{15073}a^{6}+\frac{2847}{15073}a^{5}-\frac{2013}{15073}a^{4}-\frac{7333}{15073}a^{3}+\frac{7484}{15073}a^{2}-\frac{1658}{15073}a-\frac{1032}{15073}$ 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:  $8$
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:   $\frac{28906}{15073}a^{17}-\frac{47414}{15073}a^{16}+\frac{28714}{15073}a^{15}+\frac{136700}{15073}a^{14}-\frac{247506}{15073}a^{13}+\frac{143852}{15073}a^{12}+\frac{342675}{15073}a^{11}-\frac{754909}{15073}a^{10}+\frac{654293}{15073}a^{9}+\frac{235827}{15073}a^{8}-\frac{1258657}{15073}a^{7}+\frac{1502205}{15073}a^{6}-\frac{455388}{15073}a^{5}-\frac{1151546}{15073}a^{4}+\frac{2008610}{15073}a^{3}-\frac{1668295}{15073}a^{2}+\frac{744569}{15073}a-\frac{152255}{15073}$, $\frac{110167}{15073}a^{17}-\frac{155130}{15073}a^{16}+\frac{8975}{15073}a^{15}+\frac{572942}{15073}a^{14}-\frac{761651}{15073}a^{13}+\frac{46539}{15073}a^{12}+\frac{1534879}{15073}a^{11}-\frac{2284521}{15073}a^{10}+\frac{1141987}{15073}a^{9}+\frac{1880938}{15073}a^{8}-\frac{4266984}{15073}a^{7}+\frac{3453830}{15073}a^{6}+\frac{684750}{15073}a^{5}-\frac{4745117}{15073}a^{4}+\frac{5243301}{15073}a^{3}-\frac{3048018}{15073}a^{2}+\frac{917181}{15073}a-\frac{102216}{15073}$, $a$, $\frac{62425}{15073}a^{17}-\frac{68368}{15073}a^{16}-\frac{7164}{15073}a^{15}+\frac{314046}{15073}a^{14}-\frac{339399}{15073}a^{13}-\frac{36767}{15073}a^{12}+\frac{818836}{15073}a^{11}-\frac{1061198}{15073}a^{10}+\frac{430348}{15073}a^{9}+\frac{1078872}{15073}a^{8}-\frac{2065035}{15073}a^{7}+\frac{1485979}{15073}a^{6}+\frac{601152}{15073}a^{5}-\frac{2379458}{15073}a^{4}+\frac{2370946}{15073}a^{3}-\frac{1295213}{15073}a^{2}+\frac{352320}{15073}a-\frac{30744}{15073}$, $\frac{109774}{15073}a^{17}-\frac{124768}{15073}a^{16}-\frac{23667}{15073}a^{15}+\frac{560108}{15073}a^{14}-\frac{606390}{15073}a^{13}-\frac{110285}{15073}a^{12}+\frac{1480660}{15073}a^{11}-\frac{1868966}{15073}a^{10}+\frac{658757}{15073}a^{9}+\frac{2005433}{15073}a^{8}-\frac{3674276}{15073}a^{7}+\frac{2473625}{15073}a^{6}+\frac{1269128}{15073}a^{5}-\frac{4300687}{15073}a^{4}+\frac{4055460}{15073}a^{3}-\frac{2025031}{15073}a^{2}+\frac{483519}{15073}a-\frac{28246}{15073}$, $\frac{109533}{15073}a^{17}-\frac{117962}{15073}a^{16}-\frac{5177}{15073}a^{15}+\frac{545104}{15073}a^{14}-\frac{592719}{15073}a^{13}-\frac{27918}{15073}a^{12}+\frac{1413123}{15073}a^{11}-\frac{1868688}{15073}a^{10}+\frac{835748}{15073}a^{9}+\frac{1818134}{15073}a^{8}-\frac{3600801}{15073}a^{7}+\frac{2715930}{15073}a^{6}+\frac{884461}{15073}a^{5}-\frac{4086868}{15073}a^{4}+\frac{4270194}{15073}a^{3}-\frac{2472105}{15073}a^{2}+\frac{762513}{15073}a-\frac{96067}{15073}$, $\frac{43873}{15073}a^{17}-\frac{96832}{15073}a^{16}+\frac{30342}{15073}a^{15}+\frac{241741}{15073}a^{14}-\frac{477595}{15073}a^{13}+\frac{146619}{15073}a^{12}+\frac{678412}{15073}a^{11}-\frac{1351640}{15073}a^{10}+\frac{889812}{15073}a^{9}+\frac{662301}{15073}a^{8}-\frac{2342180}{15073}a^{7}+\frac{2298572}{15073}a^{6}-\frac{229615}{15073}a^{5}-\frac{2415322}{15073}a^{4}+\frac{3223025}{15073}a^{3}-\frac{2129993}{15073}a^{2}+\frac{769587}{15073}a-\frac{118228}{15073}$, $\frac{25055}{15073}a^{17}-\frac{38854}{15073}a^{16}-\frac{6560}{15073}a^{15}+\frac{135551}{15073}a^{14}-\frac{179596}{15073}a^{13}-\frac{24519}{15073}a^{12}+\frac{372882}{15073}a^{11}-\frac{519681}{15073}a^{10}+\frac{190121}{15073}a^{9}+\frac{497446}{15073}a^{8}-\frac{998468}{15073}a^{7}+\frac{692922}{15073}a^{6}+\frac{307609}{15073}a^{5}-\frac{1147005}{15073}a^{4}+\frac{1111884}{15073}a^{3}-\frac{524055}{15073}a^{2}+\frac{120582}{15073}a+\frac{8508}{15073}$ 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:  \( 27.3003078572 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 27.3003078572 \cdot 1}{2\cdot\sqrt{1678237502892756571}}\cr\approx \mathstrut & 0.160816147175 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + x^16 + 5*x^15 - 10*x^14 + 5*x^13 + 13*x^12 - 29*x^11 + 24*x^10 + 9*x^9 - 48*x^8 + 56*x^7 - 16*x^6 - 44*x^5 + 74*x^4 - 60*x^3 + 29*x^2 - 8*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 - 2*x^17 + x^16 + 5*x^15 - 10*x^14 + 5*x^13 + 13*x^12 - 29*x^11 + 24*x^10 + 9*x^9 - 48*x^8 + 56*x^7 - 16*x^6 - 44*x^5 + 74*x^4 - 60*x^3 + 29*x^2 - 8*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 - 2*x^17 + x^16 + 5*x^15 - 10*x^14 + 5*x^13 + 13*x^12 - 29*x^11 + 24*x^10 + 9*x^9 - 48*x^8 + 56*x^7 - 16*x^6 - 44*x^5 + 74*x^4 - 60*x^3 + 29*x^2 - 8*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 - 2*x^17 + x^16 + 5*x^15 - 10*x^14 + 5*x^13 + 13*x^12 - 29*x^11 + 24*x^10 + 9*x^9 - 48*x^8 + 56*x^7 - 16*x^6 - 44*x^5 + 74*x^4 - 60*x^3 + 29*x^2 - 8*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 $C_2^9.S_9$ are not computed
Character table for $C_2^9.S_9$ is not computed

Intermediate fields

9.3.109880167.1

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 $18$ ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.9.0.1}{9} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ $18$ ${\href{/padicField/23.7.0.1}{7} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $16{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $18$ ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(139\) Copy content Toggle raw display 139.2.1.1$x^{2} + 278$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.0.1$x^{2} + 138 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} + 138 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.12.0.1$x^{12} + 120 x^{7} + 75 x^{6} + 41 x^{5} + 77 x^{4} + 106 x^{3} + 8 x^{2} + 10 x + 2$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(367\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $12$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(299401\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$