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Properties

Label 18.0.396...104.1
Degree $18$
Signature $[0, 9]$
Discriminant $-3.965\times 10^{29}$
Root discriminant \(44.09\)
Ramified primes $2,3$
Class number $542$ (GRH)
Class group [542] (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 36*x^16 + 540*x^14 + 4368*x^12 + 20592*x^10 + 57024*x^8 + 88704*x^6 + 69120*x^4 + 20736*x^2 + 1536)
 
gp: K = bnfinit(y^18 + 36*y^16 + 540*y^14 + 4368*y^12 + 20592*y^10 + 57024*y^8 + 88704*y^6 + 69120*y^4 + 20736*y^2 + 1536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 36*x^16 + 540*x^14 + 4368*x^12 + 20592*x^10 + 57024*x^8 + 88704*x^6 + 69120*x^4 + 20736*x^2 + 1536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 36*x^16 + 540*x^14 + 4368*x^12 + 20592*x^10 + 57024*x^8 + 88704*x^6 + 69120*x^4 + 20736*x^2 + 1536)
 

\( x^{18} + 36 x^{16} + 540 x^{14} + 4368 x^{12} + 20592 x^{10} + 57024 x^{8} + 88704 x^{6} + 69120 x^{4} + \cdots + 1536 \) 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:   \(-396521139274783615537700143104\) \(\medspace = -\,2^{27}\cdot 3^{45}\) 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:  \(44.09\)
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:  $2^{3/2}3^{5/2}\approx 44.090815370097204$
Ramified primes:   \(2\), \(3\) 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:  \(\Q(\sqrt{-6}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(216=2^{3}\cdot 3^{3}\)
Dirichlet character group:    $\lbrace$$\chi_{216}(1,·)$, $\chi_{216}(197,·)$, $\chi_{216}(193,·)$, $\chi_{216}(73,·)$, $\chi_{216}(77,·)$, $\chi_{216}(145,·)$, $\chi_{216}(149,·)$, $\chi_{216}(25,·)$, $\chi_{216}(29,·)$, $\chi_{216}(5,·)$, $\chi_{216}(97,·)$, $\chi_{216}(101,·)$, $\chi_{216}(169,·)$, $\chi_{216}(173,·)$, $\chi_{216}(49,·)$, $\chi_{216}(53,·)$, $\chi_{216}(121,·)$, $\chi_{216}(125,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{256}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$ 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

$C_{542}$, which has order $542$ (assuming GRH)

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:   $\frac{1}{8}a^{6}+\frac{3}{2}a^{4}+\frac{9}{2}a^{2}+1$, $\frac{1}{8}a^{6}+\frac{3}{2}a^{4}+\frac{9}{2}a^{2}+2$, $\frac{1}{16}a^{8}+\frac{7}{8}a^{6}+\frac{15}{4}a^{4}+5a^{2}+1$, $\frac{1}{16}a^{8}+a^{6}+5a^{4}+8a^{2}+2$, $\frac{1}{4}a^{4}+2a^{2}+2$, $\frac{1}{128}a^{14}+\frac{15}{64}a^{12}+\frac{45}{16}a^{10}+\frac{275}{16}a^{8}+\frac{449}{8}a^{6}+\frac{371}{4}a^{4}+\frac{125}{2}a^{2}+5$, $\frac{1}{2}a^{2}+2$, $\frac{1}{128}a^{14}+\frac{7}{32}a^{12}+\frac{77}{32}a^{10}+\frac{105}{8}a^{8}+\frac{147}{4}a^{6}+49a^{4}+\frac{49}{2}a^{2}+2$ Copy content Toggle raw display (assuming GRH)
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:  \( 40934.0329443 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 40934.0329443 \cdot 542}{2\cdot\sqrt{396521139274783615537700143104}}\cr\approx \mathstrut & 0.268868217638 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 + 36*x^16 + 540*x^14 + 4368*x^12 + 20592*x^10 + 57024*x^8 + 88704*x^6 + 69120*x^4 + 20736*x^2 + 1536)
 
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 + 36*x^16 + 540*x^14 + 4368*x^12 + 20592*x^10 + 57024*x^8 + 88704*x^6 + 69120*x^4 + 20736*x^2 + 1536, 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 + 36*x^16 + 540*x^14 + 4368*x^12 + 20592*x^10 + 57024*x^8 + 88704*x^6 + 69120*x^4 + 20736*x^2 + 1536);
 
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 + 36*x^16 + 540*x^14 + 4368*x^12 + 20592*x^10 + 57024*x^8 + 88704*x^6 + 69120*x^4 + 20736*x^2 + 1536);
 
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

$C_{18}$ (as 18T1):

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 cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\zeta_{9})^+\), 6.0.10077696.1, \(\Q(\zeta_{27})^+\)

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.9.0.1}{9} }^{2}$ ${\href{/padicField/7.9.0.1}{9} }^{2}$ ${\href{/padicField/11.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ $18$ ${\href{/padicField/29.9.0.1}{9} }^{2}$ ${\href{/padicField/31.9.0.1}{9} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ $18$ $18$ $18$ ${\href{/padicField/53.1.0.1}{1} }^{18}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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
\(2\) Copy content Toggle raw display 2.18.27.73$x^{18} + 278 x^{16} + 1088 x^{15} + 26288 x^{14} + 209856 x^{13} + 1474592 x^{12} + 10704128 x^{11} + 72384608 x^{10} + 143870720 x^{9} - 153804736 x^{8} - 1910555648 x^{7} - 2894992640 x^{6} + 10517648384 x^{5} + 52165493248 x^{4} + 98302275584 x^{3} + 92899287296 x^{2} + 41262387200 x + 7115101696$$2$$9$$27$$C_{18}$$[3]^{9}$
\(3\) Copy content Toggle raw display Deg $18$$18$$1$$45$