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Properties

Label 18.0.134...623.1
Degree $18$
Signature $[0, 9]$
Discriminant $-1.342\times 10^{19}$
Root discriminant \(11.55\)
Ramified primes $7,41$
Class number $1$
Class group trivial
Galois group $C_3^3:C_6$ (as 18T85)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 23*x^15 + 37*x^14 - 78*x^13 + 152*x^12 - 240*x^11 + 302*x^10 - 308*x^9 + 300*x^8 - 298*x^7 + 220*x^6 - 85*x^5 + 25*x^4 - 26*x^3 + 12*x^2 + 1)
 
gp: K = bnfinit(y^18 - 6*y^17 + 15*y^16 - 23*y^15 + 37*y^14 - 78*y^13 + 152*y^12 - 240*y^11 + 302*y^10 - 308*y^9 + 300*y^8 - 298*y^7 + 220*y^6 - 85*y^5 + 25*y^4 - 26*y^3 + 12*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 15*x^16 - 23*x^15 + 37*x^14 - 78*x^13 + 152*x^12 - 240*x^11 + 302*x^10 - 308*x^9 + 300*x^8 - 298*x^7 + 220*x^6 - 85*x^5 + 25*x^4 - 26*x^3 + 12*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 15*x^16 - 23*x^15 + 37*x^14 - 78*x^13 + 152*x^12 - 240*x^11 + 302*x^10 - 308*x^9 + 300*x^8 - 298*x^7 + 220*x^6 - 85*x^5 + 25*x^4 - 26*x^3 + 12*x^2 + 1)
 

\( x^{18} - 6 x^{17} + 15 x^{16} - 23 x^{15} + 37 x^{14} - 78 x^{13} + 152 x^{12} - 240 x^{11} + 302 x^{10} + \cdots + 1 \) 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:   \(-13415474159898041623\) \(\medspace = -\,7^{15}\cdot 41^{4}\) 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:  \(11.55\)
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:  $7^{5/6}41^{2/3}\approx 60.17797984027825$
Ramified primes:   \(7\), \(41\) 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:  \(\Q(\sqrt{-7}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{2}{7}a^{11}-\frac{2}{7}a^{10}+\frac{2}{7}a^{9}+\frac{1}{7}a^{8}+\frac{1}{7}a^{6}+\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-\frac{2}{7}a^{2}+\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{13}+\frac{1}{7}a^{11}-\frac{2}{7}a^{10}-\frac{2}{7}a^{9}+\frac{2}{7}a^{8}+\frac{1}{7}a^{7}+\frac{2}{7}a^{6}+\frac{1}{7}a^{5}+\frac{1}{7}a^{3}-\frac{2}{7}a^{2}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{14}+\frac{2}{7}a^{7}-\frac{1}{7}$, $\frac{1}{7}a^{15}+\frac{2}{7}a^{8}-\frac{1}{7}a$, $\frac{1}{91}a^{16}-\frac{2}{91}a^{15}-\frac{4}{91}a^{14}-\frac{4}{91}a^{13}+\frac{31}{91}a^{11}+\frac{29}{91}a^{10}+\frac{3}{91}a^{9}-\frac{5}{91}a^{8}+\frac{16}{91}a^{7}-\frac{36}{91}a^{6}+\frac{45}{91}a^{5}+\frac{6}{13}a^{4}+\frac{17}{91}a^{3}+\frac{3}{13}a^{2}-\frac{25}{91}a+\frac{45}{91}$, $\frac{1}{294931}a^{17}-\frac{899}{294931}a^{16}-\frac{17151}{294931}a^{15}+\frac{18300}{294931}a^{14}-\frac{57}{22687}a^{13}-\frac{18975}{294931}a^{12}-\frac{119038}{294931}a^{11}-\frac{140228}{294931}a^{10}-\frac{298}{637}a^{9}-\frac{129022}{294931}a^{8}-\frac{3845}{294931}a^{7}+\frac{88575}{294931}a^{6}+\frac{54447}{294931}a^{5}-\frac{116450}{294931}a^{4}-\frac{10574}{294931}a^{3}+\frac{11246}{294931}a^{2}+\frac{40085}{294931}a-\frac{6659}{22687}$ 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:   \( \frac{741}{3241} a^{17} - \frac{39470}{42133} a^{16} + \frac{39303}{42133} a^{15} + \frac{2564}{6019} a^{14} + \frac{31039}{42133} a^{13} - \frac{2328}{463} a^{12} + \frac{240360}{42133} a^{11} + \frac{274}{6019} a^{10} - \frac{1200}{91} a^{9} + \frac{1119011}{42133} a^{8} - \frac{932756}{42133} a^{7} + \frac{836362}{42133} a^{6} - \frac{1636543}{42133} a^{5} + \frac{194663}{6019} a^{4} - \frac{7777}{6019} a^{3} - \frac{76417}{42133} a^{2} - \frac{26186}{6019} a - \frac{13298}{42133} \)  (order $14$) 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:   $\frac{156918}{294931}a^{17}-\frac{876586}{294931}a^{16}+\frac{1934731}{294931}a^{15}-\frac{2652281}{294931}a^{14}+\frac{4664918}{294931}a^{13}-\frac{10386791}{294931}a^{12}+\frac{19164477}{294931}a^{11}-\frac{28939879}{294931}a^{10}+\frac{75149}{637}a^{9}-\frac{34123126}{294931}a^{8}+\frac{35055988}{294931}a^{7}-\frac{34996763}{294931}a^{6}+\frac{21316868}{294931}a^{5}-\frac{7807123}{294931}a^{4}+\frac{5741516}{294931}a^{3}-\frac{3281877}{294931}a^{2}+\frac{80798}{294931}a-\frac{649672}{294931}$, $\frac{115119}{294931}a^{17}-\frac{690702}{294931}a^{16}+\frac{1680346}{294931}a^{15}-\frac{188091}{22687}a^{14}+\frac{3947479}{294931}a^{13}-\frac{8634905}{294931}a^{12}+\frac{16638393}{294931}a^{11}-\frac{1979864}{22687}a^{10}+\frac{68493}{637}a^{9}-\frac{31236407}{294931}a^{8}+\frac{2337301}{22687}a^{7}-\frac{30750851}{294931}a^{6}+\frac{20973369}{294931}a^{5}-\frac{6396347}{294931}a^{4}+\frac{2300548}{294931}a^{3}-\frac{2251394}{294931}a^{2}+\frac{430645}{294931}a-\frac{60227}{294931}$, $\frac{28540}{294931}a^{17}-\frac{27632}{294931}a^{16}-\frac{20544}{22687}a^{15}+\frac{581471}{294931}a^{14}-\frac{302028}{294931}a^{13}+\frac{620944}{294931}a^{12}-\frac{2497491}{294931}a^{11}+\frac{4822205}{294931}a^{10}-\frac{15810}{637}a^{9}+\frac{572675}{22687}a^{8}-\frac{4701972}{294931}a^{7}+\frac{514503}{22687}a^{6}-\frac{8085811}{294931}a^{5}+\frac{723193}{294931}a^{4}+\frac{177048}{22687}a^{3}+\frac{1485747}{294931}a^{2}-\frac{925411}{294931}a-\frac{370872}{294931}$, $\frac{15674}{294931}a^{17}-\frac{177313}{294931}a^{16}+\frac{596315}{294931}a^{15}-\frac{888816}{294931}a^{14}+\frac{1028707}{294931}a^{13}-\frac{2398884}{294931}a^{12}+\frac{5373533}{294931}a^{11}-\frac{8547424}{294931}a^{10}+\frac{22536}{637}a^{9}-\frac{9014038}{294931}a^{8}+\frac{6585577}{294931}a^{7}-\frac{7554704}{294931}a^{6}+\frac{5618257}{294931}a^{5}+\frac{2057946}{294931}a^{4}-\frac{3528067}{294931}a^{3}-\frac{400347}{294931}a^{2}+\frac{730978}{294931}a+\frac{479642}{294931}$, $\frac{12927}{42133}a^{17}-\frac{57485}{42133}a^{16}+\frac{98433}{42133}a^{15}-\frac{16365}{6019}a^{14}+\frac{33218}{6019}a^{13}-\frac{527190}{42133}a^{12}+\frac{923109}{42133}a^{11}-\frac{1244949}{42133}a^{10}+\frac{366}{13}a^{9}-\frac{864137}{42133}a^{8}+\frac{115900}{6019}a^{7}-\frac{583078}{42133}a^{6}-\frac{26527}{6019}a^{5}+\frac{596779}{42133}a^{4}-\frac{49739}{6019}a^{3}+\frac{173760}{42133}a^{2}-\frac{91830}{42133}a+\frac{3186}{42133}$, $\frac{136770}{294931}a^{17}-\frac{582552}{294931}a^{16}+\frac{1026938}{294931}a^{15}-\frac{1445905}{294931}a^{14}+\frac{2981220}{294931}a^{13}-\frac{6095767}{294931}a^{12}+\frac{10752814}{294931}a^{11}-\frac{15699231}{294931}a^{10}+\frac{38148}{637}a^{9}-\frac{17525507}{294931}a^{8}+\frac{18578181}{294931}a^{7}-\frac{15418947}{294931}a^{6}+\frac{8546647}{294931}a^{5}-\frac{3989068}{294931}a^{4}+\frac{1519551}{294931}a^{3}+\frac{79424}{294931}a^{2}+\frac{436000}{294931}a-\frac{152515}{294931}$, $\frac{159195}{294931}a^{17}-\frac{810477}{294931}a^{16}+\frac{1462768}{294931}a^{15}-\frac{1414656}{294931}a^{14}+\frac{2825334}{294931}a^{13}-\frac{7541497}{294931}a^{12}+\frac{12953266}{294931}a^{11}-\frac{16675012}{294931}a^{10}+\frac{33281}{637}a^{9}-\frac{9708081}{294931}a^{8}+\frac{10646893}{294931}a^{7}-\frac{11598898}{294931}a^{6}-\frac{2597384}{294931}a^{5}+\frac{9304640}{294931}a^{4}+\frac{439631}{294931}a^{3}-\frac{145499}{22687}a^{2}-\frac{1921637}{294931}a-\frac{372867}{294931}$, $\frac{77771}{294931}a^{17}-\frac{351305}{294931}a^{16}+\frac{621606}{294931}a^{15}-\frac{814798}{294931}a^{14}+\frac{1766255}{294931}a^{13}-\frac{3788370}{294931}a^{12}+\frac{6473814}{294931}a^{11}-\frac{9436270}{294931}a^{10}+\frac{1796}{49}a^{9}-\frac{10956132}{294931}a^{8}+\frac{12722295}{294931}a^{7}-\frac{11587495}{294931}a^{6}+\frac{503005}{22687}a^{5}-\frac{4358842}{294931}a^{4}+\frac{3511343}{294931}a^{3}-\frac{1011545}{294931}a^{2}+\frac{496569}{294931}a-\frac{203439}{294931}$ 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:  \( 765.05787444 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 765.05787444 \cdot 1}{14\cdot\sqrt{13415474159898041623}}\cr\approx \mathstrut & 0.22770997496 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 23*x^15 + 37*x^14 - 78*x^13 + 152*x^12 - 240*x^11 + 302*x^10 - 308*x^9 + 300*x^8 - 298*x^7 + 220*x^6 - 85*x^5 + 25*x^4 - 26*x^3 + 12*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^18 - 6*x^17 + 15*x^16 - 23*x^15 + 37*x^14 - 78*x^13 + 152*x^12 - 240*x^11 + 302*x^10 - 308*x^9 + 300*x^8 - 298*x^7 + 220*x^6 - 85*x^5 + 25*x^4 - 26*x^3 + 12*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^18 - 6*x^17 + 15*x^16 - 23*x^15 + 37*x^14 - 78*x^13 + 152*x^12 - 240*x^11 + 302*x^10 - 308*x^9 + 300*x^8 - 298*x^7 + 220*x^6 - 85*x^5 + 25*x^4 - 26*x^3 + 12*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^18 - 6*x^17 + 15*x^16 - 23*x^15 + 37*x^14 - 78*x^13 + 152*x^12 - 240*x^11 + 302*x^10 - 308*x^9 + 300*x^8 - 298*x^7 + 220*x^6 - 85*x^5 + 25*x^4 - 26*x^3 + 12*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

$C_3^3:C_6$ (as 18T85):

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 162
The 13 conjugacy class representatives for $C_3^3:C_6$
Character table for $C_3^3:C_6$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.3.1384375783.1 x3

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 siblings: data not computed
Minimal sibling: 9.3.1384375783.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.9.0.1}{9} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }^{3}$ ${\href{/padicField/5.6.0.1}{6} }^{3}$ R ${\href{/padicField/11.9.0.1}{9} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{9}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.9.0.1}{9} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ R ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.9.0.1}{9} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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
\(7\) Copy content Toggle raw display 7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
\(41\) Copy content Toggle raw display 41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
41.6.4.1$x^{6} + 114 x^{5} + 4350 x^{4} + 56322 x^{3} + 30774 x^{2} + 180240 x + 2223605$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$