Properties

Label 18.0.114...403.1
Degree $18$
Signature $[0, 9]$
Discriminant $-1.144\times 10^{19}$
Root discriminant \(11.45\)
Ramified primes $23,43,347$
Class number $1$
Class group trivial
Galois group $D_6\wr S_3$ (as 18T556)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 7*x^16 - 9*x^15 + 15*x^14 - 24*x^13 + 28*x^12 - 30*x^11 + 36*x^10 - 39*x^9 + 36*x^8 - 30*x^7 + 28*x^6 - 24*x^5 + 15*x^4 - 9*x^3 + 7*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^18 - 4*y^17 + 7*y^16 - 9*y^15 + 15*y^14 - 24*y^13 + 28*y^12 - 30*y^11 + 36*y^10 - 39*y^9 + 36*y^8 - 30*y^7 + 28*y^6 - 24*y^5 + 15*y^4 - 9*y^3 + 7*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^17 + 7*x^16 - 9*x^15 + 15*x^14 - 24*x^13 + 28*x^12 - 30*x^11 + 36*x^10 - 39*x^9 + 36*x^8 - 30*x^7 + 28*x^6 - 24*x^5 + 15*x^4 - 9*x^3 + 7*x^2 - 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 7*x^16 - 9*x^15 + 15*x^14 - 24*x^13 + 28*x^12 - 30*x^11 + 36*x^10 - 39*x^9 + 36*x^8 - 30*x^7 + 28*x^6 - 24*x^5 + 15*x^4 - 9*x^3 + 7*x^2 - 4*x + 1)
 

\( x^{18} - 4 x^{17} + 7 x^{16} - 9 x^{15} + 15 x^{14} - 24 x^{13} + 28 x^{12} - 30 x^{11} + 36 x^{10} - 39 x^{9} + 36 x^{8} - 30 x^{7} + 28 x^{6} - 24 x^{5} + 15 x^{4} - 9 x^{3} + 7 x^{2} - 4 x + 1 \) 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:   \(-11436479389438477403\) \(\medspace = -\,23^{6}\cdot 43^{2}\cdot 347^{3}\) 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:  \(11.45\)
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:  $23^{1/2}43^{1/2}347^{1/2}\approx 585.8182311946258$
Ramified primes:   \(23\), \(43\), \(347\) 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:  \(\Q(\sqrt{-347}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Yes
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:   $a^{17}-2a^{16}+2a^{14}+a^{13}-7a^{11}+8a^{10}-5a^{9}+7a^{8}-8a^{7}+9a^{6}+2a^{4}-3a^{3}-a^{2}+2a-1$, $a$, $a^{15}-2a^{14}+2a^{12}+a^{11}-7a^{9}+8a^{8}-6a^{7}+10a^{6}-12a^{5}+13a^{4}-5a^{3}+7a^{2}-7a+3$, $a^{17}-2a^{16}+3a^{14}-3a^{13}+7a^{12}-15a^{11}+19a^{10}-22a^{9}+27a^{8}-27a^{7}+29a^{6}-21a^{5}+20a^{4}-14a^{3}+9a^{2}-5a+2$, $2a^{17}-9a^{16}+18a^{15}-24a^{14}+33a^{13}-48a^{12}+57a^{11}-57a^{10}+57a^{9}-56a^{8}+49a^{7}-33a^{6}+23a^{5}-19a^{4}+9a^{3}-2a^{2}-a$, $2a^{17}-9a^{16}+18a^{15}-23a^{14}+29a^{13}-41a^{12}+49a^{11}-46a^{10}+40a^{9}-36a^{8}+30a^{7}-14a^{6}+4a^{5}-2a^{4}-2a^{3}+6a^{2}-7a+3$, $a^{17}-2a^{16}+3a^{14}-3a^{13}+7a^{12}-15a^{11}+19a^{10}-22a^{9}+27a^{8}-27a^{7}+29a^{6}-21a^{5}+20a^{4}-14a^{3}+9a^{2}-5a+3$, $3a^{17}-11a^{16}+15a^{15}-12a^{14}+21a^{13}-37a^{12}+32a^{11}-24a^{10}+35a^{9}-37a^{8}+25a^{7}-14a^{6}+20a^{5}-22a^{4}+4a^{3}-2a^{2}+7a-4$ 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:  \( 88.267766218 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 88.267766218 \cdot 1}{2\cdot\sqrt{11436479389438477403}}\cr\approx \mathstrut & 0.19917956512 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 7*x^16 - 9*x^15 + 15*x^14 - 24*x^13 + 28*x^12 - 30*x^11 + 36*x^10 - 39*x^9 + 36*x^8 - 30*x^7 + 28*x^6 - 24*x^5 + 15*x^4 - 9*x^3 + 7*x^2 - 4*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 - 4*x^17 + 7*x^16 - 9*x^15 + 15*x^14 - 24*x^13 + 28*x^12 - 30*x^11 + 36*x^10 - 39*x^9 + 36*x^8 - 30*x^7 + 28*x^6 - 24*x^5 + 15*x^4 - 9*x^3 + 7*x^2 - 4*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 - 4*x^17 + 7*x^16 - 9*x^15 + 15*x^14 - 24*x^13 + 28*x^12 - 30*x^11 + 36*x^10 - 39*x^9 + 36*x^8 - 30*x^7 + 28*x^6 - 24*x^5 + 15*x^4 - 9*x^3 + 7*x^2 - 4*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 - 4*x^17 + 7*x^16 - 9*x^15 + 15*x^14 - 24*x^13 + 28*x^12 - 30*x^11 + 36*x^10 - 39*x^9 + 36*x^8 - 30*x^7 + 28*x^6 - 24*x^5 + 15*x^4 - 9*x^3 + 7*x^2 - 4*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

$D_6\wr S_3$ (as 18T556):

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 10368
The 98 conjugacy class representatives for $D_6\wr S_3$ are not computed
Character table for $D_6\wr S_3$ is not computed

Intermediate fields

3.1.23.1, 6.0.183563.3, 9.3.181543807.1

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 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 18.0.1417200615982289707.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $18$ ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ R ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ R ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(23\) Copy content Toggle raw display 23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.6.3.2$x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(43\) Copy content Toggle raw display 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}$
43.4.0.1$x^{4} + 5 x^{2} + 42 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.0.1$x^{4} + 5 x^{2} + 42 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.0.1$x^{4} + 5 x^{2} + 42 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
\(347\) Copy content Toggle raw display $\Q_{347}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{347}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$