Properties

Label 18.0.61074532263...9264.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 13^{12}$
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 for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 8, 0, 22, 0, 39, 0, 57, 0, 55, 0, 29, 0, 12, 0, 5, 0, 1]);
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(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)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut +\mathstrut 5 x^{16} \) \(\mathstrut +\mathstrut 12 x^{14} \) \(\mathstrut +\mathstrut 29 x^{12} \) \(\mathstrut +\mathstrut 55 x^{10} \) \(\mathstrut +\mathstrut 57 x^{8} \) \(\mathstrut +\mathstrut 39 x^{6} \) \(\mathstrut +\mathstrut 22 x^{4} \) \(\mathstrut +\mathstrut 8 x^{2} \) \(\mathstrut +\mathstrut 1 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $18$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 9]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-6107453226347659264=-\,2^{18}\cdot 13^{12}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.06$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 13$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is not a CM field.

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

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

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

Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $8$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
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 $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{9}{94} a^{17} + \frac{56}{47} a^{16} + \frac{30}{47} a^{15} + \frac{945}{188} a^{14} + \frac{161}{94} a^{13} + \frac{490}{47} a^{12} + \frac{357}{94} a^{11} + \frac{1250}{47} a^{10} + \frac{761}{94} a^{9} + \frac{2124}{47} a^{8} + \frac{452}{47} a^{7} + \frac{3171}{94} a^{6} + \frac{263}{47} a^{5} + \frac{2039}{94} a^{4} + \frac{177}{47} a^{3} + \frac{495}{47} a^{2} + \frac{96}{47} a + \frac{251}{188} \),  \( \frac{199}{188} a^{17} - \frac{13}{47} a^{16} + \frac{841}{188} a^{15} - \frac{95}{94} a^{14} + \frac{853}{94} a^{13} - \frac{157}{94} a^{12} + \frac{2153}{94} a^{11} - \frac{218}{47} a^{10} + \frac{1824}{47} a^{9} - \frac{637}{94} a^{8} + \frac{2469}{94} a^{7} - \frac{42}{47} a^{6} + \frac{649}{47} a^{5} - \frac{13}{47} a^{4} + \frac{1263}{188} a^{3} - \frac{10}{47} a^{2} + \frac{125}{188} a + \frac{36}{47} \),  \( \frac{199}{188} a^{17} + \frac{13}{47} a^{16} + \frac{841}{188} a^{15} + \frac{95}{94} a^{14} + \frac{853}{94} a^{13} + \frac{157}{94} a^{12} + \frac{2153}{94} a^{11} + \frac{218}{47} a^{10} + \frac{1824}{47} a^{9} + \frac{637}{94} a^{8} + \frac{2469}{94} a^{7} + \frac{42}{47} a^{6} + \frac{649}{47} a^{5} + \frac{13}{47} a^{4} + \frac{1263}{188} a^{3} + \frac{10}{47} a^{2} + \frac{125}{188} a - \frac{36}{47} \),  \( \frac{9}{94} a^{17} - \frac{56}{47} a^{16} + \frac{30}{47} a^{15} - \frac{945}{188} a^{14} + \frac{161}{94} a^{13} - \frac{490}{47} a^{12} + \frac{357}{94} a^{11} - \frac{1250}{47} a^{10} + \frac{761}{94} a^{9} - \frac{2124}{47} a^{8} + \frac{452}{47} a^{7} - \frac{3171}{94} a^{6} + \frac{263}{47} a^{5} - \frac{2039}{94} a^{4} + \frac{177}{47} a^{3} - \frac{495}{47} a^{2} + \frac{96}{47} a - \frac{251}{188} \),  \( \frac{12}{47} a^{16} + \frac{33}{47} a^{14} + \frac{69}{94} a^{12} + \frac{247}{94} a^{10} + \frac{165}{94} a^{8} - \frac{519}{94} a^{6} - \frac{399}{94} a^{4} - \frac{325}{94} a^{2} - \frac{193}{94} \),  \( \frac{1}{47} a^{17} + \frac{29}{94} a^{15} + \frac{44}{47} a^{13} + \frac{71}{47} a^{11} + \frac{189}{47} a^{9} + \frac{231}{47} a^{7} - \frac{46}{47} a^{5} + \frac{8}{47} a^{3} + \frac{121}{94} a \),  \( \frac{175}{94} a^{16} + \frac{775}{94} a^{14} + \frac{1637}{94} a^{12} + \frac{4059}{94} a^{10} + \frac{7131}{94} a^{8} + \frac{5457}{94} a^{6} + \frac{2995}{94} a^{4} + \frac{794}{47} a^{2} + \frac{112}{47} \),  \( \frac{247}{188} a^{17} - \frac{79}{188} a^{16} + \frac{255}{47} a^{15} - \frac{185}{94} a^{14} + \frac{2079}{188} a^{13} - \frac{211}{47} a^{12} + \frac{5411}{188} a^{11} - \frac{521}{47} a^{10} + \frac{9083}{188} a^{9} - \frac{948}{47} a^{8} + \frac{6579}{188} a^{7} - \frac{1769}{94} a^{6} + \frac{4853}{188} a^{5} - \frac{1191}{94} a^{4} + \frac{1317}{94} a^{3} - \frac{1055}{188} a^{2} + \frac{303}{188} a - \frac{14}{47} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 132.794513852 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

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

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
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.

Sibling fields

Degree 6 sibling: 6.0.10816.1
Degree 9 sibling: data not computed

Frobenius cycle types

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

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$