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Properties

Label 18.0.610...264.1
Degree $18$
Signature $[0, 9]$
Discriminant $-6.107\times 10^{18}$
Root discriminant \(11.06\)
Ramified primes $2,13$
Class number $1$
Class group trivial
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 5*x^16 + 12*x^14 + 29*x^12 + 55*x^10 + 57*x^8 + 39*x^6 + 22*x^4 + 8*x^2 + 1)
 
gp: K = bnfinit(y^18 + 5*y^16 + 12*y^14 + 29*y^12 + 55*y^10 + 57*y^8 + 39*y^6 + 22*y^4 + 8*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 5*x^16 + 12*x^14 + 29*x^12 + 55*x^10 + 57*x^8 + 39*x^6 + 22*x^4 + 8*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 5*x^16 + 12*x^14 + 29*x^12 + 55*x^10 + 57*x^8 + 39*x^6 + 22*x^4 + 8*x^2 + 1)
 

\( x^{18} + 5x^{16} + 12x^{14} + 29x^{12} + 55x^{10} + 57x^{8} + 39x^{6} + 22x^{4} + 8x^{2} + 1 \) 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:   \(-6107453226347659264\) \(\medspace = -\,2^{18}\cdot 13^{12}\) 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:  \(11.06\)
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:   \(2\), \(13\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{11}+\frac{1}{4}a^{10}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{13}+\frac{1}{4}a^{11}-\frac{1}{2}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{188}a^{16}-\frac{9}{188}a^{14}-\frac{3}{188}a^{12}-\frac{1}{2}a^{11}-\frac{23}{188}a^{10}-\frac{1}{2}a^{9}-\frac{93}{188}a^{8}-\frac{1}{2}a^{7}-\frac{51}{188}a^{6}+\frac{1}{188}a^{4}+\frac{2}{47}a^{2}-\frac{1}{2}a-\frac{5}{94}$, $\frac{1}{188}a^{17}-\frac{9}{188}a^{15}-\frac{3}{188}a^{13}-\frac{23}{188}a^{11}-\frac{93}{188}a^{9}-\frac{51}{188}a^{7}-\frac{1}{2}a^{6}+\frac{1}{188}a^{5}-\frac{1}{2}a^{4}+\frac{2}{47}a^{3}-\frac{5}{94}a-\frac{1}{2}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

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:   \( -\frac{74}{47} a^{17} - \frac{689}{94} a^{15} - \frac{765}{47} a^{13} - \frac{1870}{47} a^{11} - \frac{3411}{47} a^{9} - \frac{2994}{47} a^{7} - \frac{1813}{47} a^{5} - \frac{1062}{47} a^{3} - \frac{541}{94} a \)  (order $4$) 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:   $\frac{9}{94}a^{17}+\frac{56}{47}a^{16}+\frac{30}{47}a^{15}+\frac{945}{188}a^{14}+\frac{161}{94}a^{13}+\frac{490}{47}a^{12}+\frac{357}{94}a^{11}+\frac{1250}{47}a^{10}+\frac{761}{94}a^{9}+\frac{2124}{47}a^{8}+\frac{452}{47}a^{7}+\frac{3171}{94}a^{6}+\frac{263}{47}a^{5}+\frac{2039}{94}a^{4}+\frac{177}{47}a^{3}+\frac{495}{47}a^{2}+\frac{96}{47}a+\frac{251}{188}$, $\frac{199}{188}a^{17}-\frac{13}{47}a^{16}+\frac{841}{188}a^{15}-\frac{95}{94}a^{14}+\frac{853}{94}a^{13}-\frac{157}{94}a^{12}+\frac{2153}{94}a^{11}-\frac{218}{47}a^{10}+\frac{1824}{47}a^{9}-\frac{637}{94}a^{8}+\frac{2469}{94}a^{7}-\frac{42}{47}a^{6}+\frac{649}{47}a^{5}-\frac{13}{47}a^{4}+\frac{1263}{188}a^{3}-\frac{10}{47}a^{2}+\frac{125}{188}a+\frac{36}{47}$, $\frac{199}{188}a^{17}+\frac{13}{47}a^{16}+\frac{841}{188}a^{15}+\frac{95}{94}a^{14}+\frac{853}{94}a^{13}+\frac{157}{94}a^{12}+\frac{2153}{94}a^{11}+\frac{218}{47}a^{10}+\frac{1824}{47}a^{9}+\frac{637}{94}a^{8}+\frac{2469}{94}a^{7}+\frac{42}{47}a^{6}+\frac{649}{47}a^{5}+\frac{13}{47}a^{4}+\frac{1263}{188}a^{3}+\frac{10}{47}a^{2}+\frac{125}{188}a-\frac{36}{47}$, $\frac{9}{94}a^{17}-\frac{56}{47}a^{16}+\frac{30}{47}a^{15}-\frac{945}{188}a^{14}+\frac{161}{94}a^{13}-\frac{490}{47}a^{12}+\frac{357}{94}a^{11}-\frac{1250}{47}a^{10}+\frac{761}{94}a^{9}-\frac{2124}{47}a^{8}+\frac{452}{47}a^{7}-\frac{3171}{94}a^{6}+\frac{263}{47}a^{5}-\frac{2039}{94}a^{4}+\frac{177}{47}a^{3}-\frac{495}{47}a^{2}+\frac{96}{47}a-\frac{251}{188}$, $\frac{12}{47}a^{16}+\frac{33}{47}a^{14}+\frac{69}{94}a^{12}+\frac{247}{94}a^{10}+\frac{165}{94}a^{8}-\frac{519}{94}a^{6}-\frac{399}{94}a^{4}-\frac{325}{94}a^{2}-\frac{193}{94}$, $\frac{1}{47}a^{17}+\frac{29}{94}a^{15}+\frac{44}{47}a^{13}+\frac{71}{47}a^{11}+\frac{189}{47}a^{9}+\frac{231}{47}a^{7}-\frac{46}{47}a^{5}+\frac{8}{47}a^{3}+\frac{121}{94}a$, $\frac{175}{94}a^{16}+\frac{775}{94}a^{14}+\frac{1637}{94}a^{12}+\frac{4059}{94}a^{10}+\frac{7131}{94}a^{8}+\frac{5457}{94}a^{6}+\frac{2995}{94}a^{4}+\frac{794}{47}a^{2}+\frac{112}{47}$, $\frac{247}{188}a^{17}-\frac{79}{188}a^{16}+\frac{255}{47}a^{15}-\frac{185}{94}a^{14}+\frac{2079}{188}a^{13}-\frac{211}{47}a^{12}+\frac{5411}{188}a^{11}-\frac{521}{47}a^{10}+\frac{9083}{188}a^{9}-\frac{948}{47}a^{8}+\frac{6579}{188}a^{7}-\frac{1769}{94}a^{6}+\frac{4853}{188}a^{5}-\frac{1191}{94}a^{4}+\frac{1317}{94}a^{3}-\frac{1055}{188}a^{2}+\frac{303}{188}a-\frac{14}{47}$ 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:  \( 132.794513852 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 132.794513852 \cdot 1}{4\cdot\sqrt{6107453226347659264}}\cr\approx \mathstrut & 0.205025973389 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 + 5*x^16 + 12*x^14 + 29*x^12 + 55*x^10 + 57*x^8 + 39*x^6 + 22*x^4 + 8*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 + 5*x^16 + 12*x^14 + 29*x^12 + 55*x^10 + 57*x^8 + 39*x^6 + 22*x^4 + 8*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 + 5*x^16 + 12*x^14 + 29*x^12 + 55*x^10 + 57*x^8 + 39*x^6 + 22*x^4 + 8*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 + 5*x^16 + 12*x^14 + 29*x^12 + 55*x^10 + 57*x^8 + 39*x^6 + 22*x^4 + 8*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

$C_3\times S_3$ (as 18T3):

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 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.676.1 x3, 3.3.169.1, 6.0.1827904.2, 6.0.10816.1 x2, 6.0.1827904.1, 9.3.308915776.1 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 6 sibling: 6.0.10816.1
Degree 9 sibling: 9.3.308915776.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}$ ${\href{/padicField/5.3.0.1}{3} }^{6}$ ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ R ${\href{/padicField/17.3.0.1}{3} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.2.0.1}{2} }^{9}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.3.0.1}{3} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.3.0.1}{3} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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
\(2\) Copy content Toggle raw display 2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
\(13\) Copy content Toggle raw display 13.9.6.1$x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$