Properties

Label 18.0.661...947.1
Degree $18$
Signature $[0, 9]$
Discriminant $-6.612\times 10^{19}$
Root discriminant \(12.62\)
Ramified primes $3,43$
Class number $1$
Class group trivial
Galois group $\SOPlus(4,2)$ (as 18T34)

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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 + 6*x^16 - 9*x^15 + 15*x^14 - 21*x^13 + 24*x^12 - 24*x^11 + 27*x^10 - 29*x^9 + 27*x^8 - 24*x^7 + 24*x^6 - 21*x^5 + 15*x^4 - 9*x^3 + 6*x^2 - 3*x + 1)
 
gp: K = bnfinit(y^18 - 3*y^17 + 6*y^16 - 9*y^15 + 15*y^14 - 21*y^13 + 24*y^12 - 24*y^11 + 27*y^10 - 29*y^9 + 27*y^8 - 24*y^7 + 24*y^6 - 21*y^5 + 15*y^4 - 9*y^3 + 6*y^2 - 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 6*x^16 - 9*x^15 + 15*x^14 - 21*x^13 + 24*x^12 - 24*x^11 + 27*x^10 - 29*x^9 + 27*x^8 - 24*x^7 + 24*x^6 - 21*x^5 + 15*x^4 - 9*x^3 + 6*x^2 - 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 6*x^16 - 9*x^15 + 15*x^14 - 21*x^13 + 24*x^12 - 24*x^11 + 27*x^10 - 29*x^9 + 27*x^8 - 24*x^7 + 24*x^6 - 21*x^5 + 15*x^4 - 9*x^3 + 6*x^2 - 3*x + 1)
 

\( x^{18} - 3 x^{17} + 6 x^{16} - 9 x^{15} + 15 x^{14} - 21 x^{13} + 24 x^{12} - 24 x^{11} + 27 x^{10} - 29 x^{9} + 27 x^{8} - 24 x^{7} + 24 x^{6} - 21 x^{5} + 15 x^{4} - 9 x^{3} + 6 x^{2} - 3 x + 1 \) 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:   \(-66123690216932995947\) \(\medspace = -\,3^{21}\cdot 43^{6}\) 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:  \(12.62\)
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))
 
Ramified primes:   \(3\), \(43\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{17}a^{17}-\frac{4}{17}a^{16}-\frac{7}{17}a^{15}-\frac{2}{17}a^{14}-\frac{4}{17}a^{12}-\frac{6}{17}a^{11}-\frac{1}{17}a^{10}-\frac{6}{17}a^{9}-\frac{6}{17}a^{8}-\frac{1}{17}a^{7}-\frac{6}{17}a^{6}-\frac{4}{17}a^{5}-\frac{2}{17}a^{3}-\frac{7}{17}a^{2}-\frac{4}{17}a+\frac{1}{17}$ 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:   \( -\frac{26}{17} a^{17} + \frac{87}{17} a^{16} - \frac{158}{17} a^{15} + \frac{239}{17} a^{14} - 23 a^{13} + \frac{580}{17} a^{12} - \frac{609}{17} a^{11} + \frac{604}{17} a^{10} - \frac{711}{17} a^{9} + \frac{785}{17} a^{8} - \frac{654}{17} a^{7} + \frac{581}{17} a^{6} - \frac{644}{17} a^{5} + 32 a^{4} - \frac{322}{17} a^{3} + \frac{216}{17} a^{2} - \frac{151}{17} a + \frac{76}{17} \)  (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:   $\frac{9}{17}a^{17}-\frac{2}{17}a^{16}+\frac{5}{17}a^{15}-\frac{1}{17}a^{14}+2a^{13}+\frac{15}{17}a^{12}-\frac{20}{17}a^{11}-\frac{9}{17}a^{10}+\frac{31}{17}a^{9}+\frac{48}{17}a^{8}-\frac{43}{17}a^{7}-\frac{20}{17}a^{6}-\frac{2}{17}a^{5}+4a^{4}-\frac{18}{17}a^{3}+\frac{5}{17}a^{2}-\frac{19}{17}a+\frac{26}{17}$, $\frac{28}{17}a^{17}-\frac{27}{17}a^{16}+\frac{25}{17}a^{15}-\frac{22}{17}a^{14}+6a^{13}-\frac{27}{17}a^{12}-\frac{66}{17}a^{11}-\frac{11}{17}a^{10}+\frac{104}{17}a^{9}+\frac{53}{17}a^{8}-\frac{96}{17}a^{7}-\frac{32}{17}a^{6}+\frac{41}{17}a^{5}+7a^{4}-\frac{56}{17}a^{3}+\frac{8}{17}a^{2}-\frac{10}{17}a+\frac{28}{17}$, $\frac{2}{17}a^{17}+\frac{9}{17}a^{16}-\frac{14}{17}a^{15}+\frac{13}{17}a^{14}-a^{13}+\frac{60}{17}a^{12}-\frac{63}{17}a^{11}+\frac{15}{17}a^{10}-\frac{63}{17}a^{9}+\frac{124}{17}a^{8}-\frac{70}{17}a^{7}+\frac{22}{17}a^{6}-\frac{93}{17}a^{5}+6a^{4}-\frac{21}{17}a^{3}+\frac{20}{17}a^{2}-\frac{42}{17}a+\frac{36}{17}$, $\frac{13}{17}a^{17}-\frac{1}{17}a^{16}-\frac{6}{17}a^{15}+\frac{25}{17}a^{14}+\frac{84}{17}a^{12}-\frac{129}{17}a^{11}+\frac{106}{17}a^{10}-\frac{78}{17}a^{9}+\frac{177}{17}a^{8}-\frac{166}{17}a^{7}+\frac{109}{17}a^{6}-\frac{103}{17}a^{5}+10a^{4}-\frac{94}{17}a^{3}+\frac{62}{17}a^{2}-\frac{52}{17}a+\frac{47}{17}$, $\frac{13}{17}a^{17}-\frac{18}{17}a^{16}+\frac{45}{17}a^{15}-\frac{60}{17}a^{14}+7a^{13}-\frac{120}{17}a^{12}+\frac{160}{17}a^{11}-\frac{166}{17}a^{10}+\frac{177}{17}a^{9}-\frac{163}{17}a^{8}+\frac{191}{17}a^{7}-\frac{146}{17}a^{6}+\frac{152}{17}a^{5}-8a^{4}+\frac{110}{17}a^{3}-\frac{40}{17}a^{2}+\frac{33}{17}a-\frac{21}{17}$, $\frac{13}{17}a^{17}-\frac{18}{17}a^{16}+\frac{11}{17}a^{15}-\frac{9}{17}a^{14}+2a^{13}-\frac{1}{17}a^{12}-\frac{78}{17}a^{11}+\frac{55}{17}a^{10}-\frac{10}{17}a^{9}+\frac{92}{17}a^{8}-\frac{115}{17}a^{7}+\frac{41}{17}a^{6}-\frac{52}{17}a^{5}+7a^{4}-\frac{94}{17}a^{3}+\frac{45}{17}a^{2}-\frac{52}{17}a+\frac{30}{17}$, $\frac{1}{17}a^{17}-\frac{4}{17}a^{16}+\frac{10}{17}a^{15}-\frac{19}{17}a^{14}+a^{13}-\frac{21}{17}a^{12}+\frac{28}{17}a^{11}-\frac{35}{17}a^{10}-\frac{6}{17}a^{9}+\frac{11}{17}a^{8}+\frac{33}{17}a^{7}-\frac{23}{17}a^{6}-\frac{38}{17}a^{5}+2a^{4}+\frac{15}{17}a^{3}+\frac{10}{17}a^{2}-\frac{38}{17}a+\frac{18}{17}$, $\frac{9}{17}a^{17}-\frac{36}{17}a^{16}+\frac{56}{17}a^{15}-\frac{86}{17}a^{14}+8a^{13}-\frac{223}{17}a^{12}+\frac{201}{17}a^{11}-\frac{196}{17}a^{10}+\frac{252}{17}a^{9}-\frac{292}{17}a^{8}+\frac{195}{17}a^{7}-\frac{173}{17}a^{6}+\frac{236}{17}a^{5}-11a^{4}+\frac{67}{17}a^{3}-\frac{63}{17}a^{2}+\frac{49}{17}a-\frac{25}{17}$ 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:  \( 893.183574652 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 893.183574652 \cdot 1}{6\cdot\sqrt{66123690216932995947}}\cr\approx \mathstrut & 0.279402155017 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 9*x^15 + 15*x^14 - 21*x^13 + 24*x^12 - 24*x^11 + 27*x^10 - 29*x^9 + 27*x^8 - 24*x^7 + 24*x^6 - 21*x^5 + 15*x^4 - 9*x^3 + 6*x^2 - 3*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 + 6*x^16 - 9*x^15 + 15*x^14 - 21*x^13 + 24*x^12 - 24*x^11 + 27*x^10 - 29*x^9 + 27*x^8 - 24*x^7 + 24*x^6 - 21*x^5 + 15*x^4 - 9*x^3 + 6*x^2 - 3*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 + 6*x^16 - 9*x^15 + 15*x^14 - 21*x^13 + 24*x^12 - 24*x^11 + 27*x^10 - 29*x^9 + 27*x^8 - 24*x^7 + 24*x^6 - 21*x^5 + 15*x^4 - 9*x^3 + 6*x^2 - 3*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 + 6*x^16 - 9*x^15 + 15*x^14 - 21*x^13 + 24*x^12 - 24*x^11 + 27*x^10 - 29*x^9 + 27*x^8 - 24*x^7 + 24*x^6 - 21*x^5 + 15*x^4 - 9*x^3 + 6*x^2 - 3*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

$\SOPlus(4,2)$ (as 18T34):

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 72
The 9 conjugacy class representatives for $\SOPlus(4,2)$
Character table for $\SOPlus(4,2)$

Intermediate fields

\(\Q(\sqrt{-3}) \), 6.0.31347.1 x2, 9.3.4694808843.1

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 6 siblings: 6.4.57960603.1, 6.0.31347.1
Degree 9 sibling: 9.3.4694808843.1
Degree 12 siblings: deg 12, 12.0.126759838761.1, deg 12, deg 12, deg 12, deg 12
Degree 18 siblings: deg 18, deg 18
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 6.0.31347.1

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 ${\href{/padicField/5.6.0.1}{6} }^{3}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }$ R ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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$$21$
\(43\) Copy content Toggle raw display $\Q_{43}$$x + 40$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 40$$1$$1$$0$Trivial$[\ ]$
43.2.1.1$x^{2} + 86$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.1$x^{2} + 86$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} + 42 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.1$x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$