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Properties

Label 18.0.239...307.1
Degree $18$
Signature $[0, 9]$
Discriminant $-2.393\times 10^{20}$
Root discriminant \(13.56\)
Ramified primes $3,7$
Class number $1$
Class group trivial
Galois group $C_3^2:C_6$ (as 18T22)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 12*x^16 - 21*x^15 + 51*x^14 - 60*x^13 + 114*x^12 - 75*x^11 + 165*x^10 - 4*x^9 + 174*x^8 + 96*x^7 + 135*x^6 + 108*x^5 + 72*x^4 + 48*x^3 + 21*x^2 + 6*x + 1)
 
gp: K = bnfinit(y^18 - 3*y^17 + 12*y^16 - 21*y^15 + 51*y^14 - 60*y^13 + 114*y^12 - 75*y^11 + 165*y^10 - 4*y^9 + 174*y^8 + 96*y^7 + 135*y^6 + 108*y^5 + 72*y^4 + 48*y^3 + 21*y^2 + 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 12*x^16 - 21*x^15 + 51*x^14 - 60*x^13 + 114*x^12 - 75*x^11 + 165*x^10 - 4*x^9 + 174*x^8 + 96*x^7 + 135*x^6 + 108*x^5 + 72*x^4 + 48*x^3 + 21*x^2 + 6*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 12*x^16 - 21*x^15 + 51*x^14 - 60*x^13 + 114*x^12 - 75*x^11 + 165*x^10 - 4*x^9 + 174*x^8 + 96*x^7 + 135*x^6 + 108*x^5 + 72*x^4 + 48*x^3 + 21*x^2 + 6*x + 1)
 

\( x^{18} - 3 x^{17} + 12 x^{16} - 21 x^{15} + 51 x^{14} - 60 x^{13} + 114 x^{12} - 75 x^{11} + 165 x^{10} + \cdots + 1 \) 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:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-239338060927275176307\) \(\medspace = -\,3^{25}\cdot 7^{10}\) 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:  \(13.56\)
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:  $3^{25/18}7^{2/3}\approx 16.82934070487988$
Ramified primes:   \(3\), \(7\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{1600394}a^{17}+\frac{41817}{800197}a^{16}+\frac{374893}{1600394}a^{15}+\frac{3286}{800197}a^{14}+\frac{727273}{1600394}a^{13}+\frac{757083}{1600394}a^{12}-\frac{118911}{800197}a^{11}-\frac{110930}{800197}a^{10}+\frac{31789}{800197}a^{9}+\frac{164117}{1600394}a^{8}+\frac{237281}{800197}a^{7}-\frac{214752}{800197}a^{6}-\frac{391193}{800197}a^{5}+\frac{246049}{800197}a^{4}-\frac{532197}{1600394}a^{3}+\frac{397881}{1600394}a^{2}+\frac{290388}{800197}a+\frac{3827}{1600394}$ 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:   \( -\frac{289}{2378} a^{17} + \frac{955}{2378} a^{16} - \frac{604}{1189} a^{15} - \frac{832}{1189} a^{14} + \frac{14279}{2378} a^{13} - \frac{17033}{1189} a^{12} + \frac{37660}{1189} a^{11} - \frac{51364}{1189} a^{10} + \frac{75289}{1189} a^{9} - \frac{112369}{2378} a^{8} + \frac{172759}{2378} a^{7} - \frac{10889}{2378} a^{6} + \frac{129403}{2378} a^{5} + \frac{65363}{2378} a^{4} + \frac{33022}{1189} a^{3} + \frac{43385}{2378} a^{2} + \frac{19945}{2378} a + \frac{2856}{1189} \)  (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:   $\frac{1086095}{1600394}a^{17}-\frac{1146908}{800197}a^{16}+\frac{4587155}{800197}a^{15}-\frac{4771632}{800197}a^{14}+\frac{29114175}{1600394}a^{13}-\frac{15175527}{1600394}a^{12}+\frac{59919709}{1600394}a^{11}+\frac{3086107}{1600394}a^{10}+\frac{118777345}{1600394}a^{9}+\frac{32993464}{800197}a^{8}+\frac{181369257}{1600394}a^{7}+\frac{62763486}{800197}a^{6}+\frac{162115561}{1600394}a^{5}+\frac{55612522}{800197}a^{4}+\frac{39310081}{800197}a^{3}+\frac{44588241}{1600394}a^{2}+\frac{8111447}{800197}a+\frac{1331469}{800197}$, $\frac{792107}{800197}a^{17}-\frac{4866175}{1600394}a^{16}+\frac{9464624}{800197}a^{15}-\frac{17158615}{800197}a^{14}+\frac{41020018}{800197}a^{13}-\frac{105013071}{1600394}a^{12}+\frac{191859867}{1600394}a^{11}-\frac{154426963}{1600394}a^{10}+\frac{286031631}{1600394}a^{9}-\frac{77978523}{1600394}a^{8}+\frac{139372904}{800197}a^{7}+\frac{54077171}{1600394}a^{6}+\frac{85565549}{800197}a^{5}+\frac{87035983}{1600394}a^{4}+\frac{31621553}{800197}a^{3}+\frac{17939575}{800197}a^{2}+\frac{9360055}{1600394}a+\frac{1047450}{800197}$, $\frac{44}{1189}a^{17}+\frac{1130}{1189}a^{16}-\frac{7733}{2378}a^{15}+\frac{14509}{1189}a^{14}-\frac{25703}{1189}a^{13}+\frac{54133}{1189}a^{12}-\frac{124403}{2378}a^{11}+\frac{198263}{2378}a^{10}-\frac{113525}{2378}a^{9}+\frac{223045}{2378}a^{8}+\frac{16855}{2378}a^{7}+\frac{84209}{1189}a^{6}+\frac{96575}{2378}a^{5}+\frac{43426}{1189}a^{4}+\frac{57257}{2378}a^{3}+\frac{13007}{1189}a^{2}+\frac{3723}{1189}a+\frac{289}{2378}$, $\frac{469304}{800197}a^{17}-\frac{1492508}{800197}a^{16}+\frac{5071461}{800197}a^{15}-\frac{7695320}{800197}a^{14}+\frac{14504143}{800197}a^{13}-\frac{10193875}{800197}a^{12}+\frac{14265021}{800197}a^{11}+\frac{13050958}{800197}a^{10}+\frac{1264370}{800197}a^{9}+\frac{48214744}{800197}a^{8}-\frac{1785374}{800197}a^{7}+\frac{49690904}{800197}a^{6}+\frac{8717849}{800197}a^{5}+\frac{22709532}{800197}a^{4}+\frac{10510495}{800197}a^{3}+\frac{4375662}{800197}a^{2}+\frac{2198946}{800197}a+\frac{1184537}{800197}$, $\frac{34061}{1600394}a^{17}-\frac{822020}{800197}a^{16}+\frac{6088323}{1600394}a^{15}-\frac{10505695}{800197}a^{14}+\frac{40757171}{1600394}a^{13}-\frac{81764553}{1600394}a^{12}+\frac{52382448}{800197}a^{11}-\frac{76675408}{800197}a^{10}+\frac{57712772}{800197}a^{9}-\frac{163427293}{1600394}a^{8}+\frac{13641790}{800197}a^{7}-\frac{51279703}{800197}a^{6}-\frac{25150633}{800197}a^{5}-\frac{19793117}{800197}a^{4}-\frac{42709817}{1600394}a^{3}-\frac{11114409}{1600394}a^{2}-\frac{4330434}{800197}a-\frac{880861}{1600394}$, $\frac{1187783}{1600394}a^{17}-\frac{1106670}{800197}a^{16}+\frac{4954307}{800197}a^{15}-\frac{5107210}{800197}a^{14}+\frac{36245835}{1600394}a^{13}-\frac{22274369}{1600394}a^{12}+\frac{82235423}{1600394}a^{11}-\frac{4661325}{1600394}a^{10}+\frac{151913523}{1600394}a^{9}+\frac{47307442}{800197}a^{8}+\frac{212357117}{1600394}a^{7}+\frac{95246317}{800197}a^{6}+\frac{182439643}{1600394}a^{5}+\frac{78489151}{800197}a^{4}+\frac{43748550}{800197}a^{3}+\frac{54353019}{1600394}a^{2}+\frac{8216697}{800197}a+\frac{663389}{800197}$, $\frac{25659}{55186}a^{17}-\frac{80769}{55186}a^{16}+\frac{160861}{27593}a^{15}-\frac{284664}{27593}a^{14}+\frac{1331657}{55186}a^{13}-\frac{746796}{27593}a^{12}+\frac{1365869}{27593}a^{11}-\frac{741966}{27593}a^{10}+\frac{1790823}{27593}a^{9}+\frac{717419}{55186}a^{8}+\frac{3886071}{55186}a^{7}+\frac{3174259}{55186}a^{6}+\frac{3604273}{55186}a^{5}+\frac{3083047}{55186}a^{4}+\frac{1128769}{27593}a^{3}+\frac{1308601}{55186}a^{2}+\frac{669699}{55186}a+\frac{79532}{27593}$, $\frac{743375}{1600394}a^{17}-\frac{1481559}{1600394}a^{16}+\frac{3538032}{800197}a^{15}-\frac{4272176}{800197}a^{14}+\frac{28274357}{1600394}a^{13}-\frac{9792117}{800197}a^{12}+\frac{31105057}{800197}a^{11}-\frac{1487703}{800197}a^{10}+\frac{50942679}{800197}a^{9}+\frac{85660743}{1600394}a^{8}+\frac{144312199}{1600394}a^{7}+\frac{179312113}{1600394}a^{6}+\frac{147428811}{1600394}a^{5}+\frac{156129827}{1600394}a^{4}+\frac{45337684}{800197}a^{3}+\frac{60386631}{1600394}a^{2}+\frac{25676909}{1600394}a+\frac{2498486}{800197}$ 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:  \( 2026.39351046 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 2026.39351046 \cdot 1}{6\cdot\sqrt{239338060927275176307}}\cr\approx \mathstrut & 0.333185019566 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 12*x^16 - 21*x^15 + 51*x^14 - 60*x^13 + 114*x^12 - 75*x^11 + 165*x^10 - 4*x^9 + 174*x^8 + 96*x^7 + 135*x^6 + 108*x^5 + 72*x^4 + 48*x^3 + 21*x^2 + 6*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 - 3*x^17 + 12*x^16 - 21*x^15 + 51*x^14 - 60*x^13 + 114*x^12 - 75*x^11 + 165*x^10 - 4*x^9 + 174*x^8 + 96*x^7 + 135*x^6 + 108*x^5 + 72*x^4 + 48*x^3 + 21*x^2 + 6*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 - 3*x^17 + 12*x^16 - 21*x^15 + 51*x^14 - 60*x^13 + 114*x^12 - 75*x^11 + 165*x^10 - 4*x^9 + 174*x^8 + 96*x^7 + 135*x^6 + 108*x^5 + 72*x^4 + 48*x^3 + 21*x^2 + 6*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 - 3*x^17 + 12*x^16 - 21*x^15 + 51*x^14 - 60*x^13 + 114*x^12 - 75*x^11 + 165*x^10 - 4*x^9 + 174*x^8 + 96*x^7 + 135*x^6 + 108*x^5 + 72*x^4 + 48*x^3 + 21*x^2 + 6*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

$C_3^2:C_6$ (as 18T22):

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 54
The 10 conjugacy class representatives for $C_3^2:C_6$
Character table for $C_3^2:C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 6.0.107163.1

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 9 siblings: data not computed
Degree 18 siblings: data not computed
Degree 27 sibling: data not computed
Minimal sibling: 9.1.62523502209.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{3}$ R ${\href{/padicField/5.6.0.1}{6} }^{3}$ R ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 Deg $18$$18$$1$$25$
\(7\) Copy content Toggle raw display 7.3.0.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$