Normalized defining polynomial
\( x^{18} - 18 x^{15} + 121 x^{12} - 414 x^{9} + 2587 x^{6} - 1080 x^{3} + 729 \)
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: | \(-833176180046218838523368177664=-\,2^{24}\cdot 3^{21}\cdot 7^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $45.95$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{9} a^{8} + \frac{2}{9} a^{2}$, $\frac{1}{9} a^{9} + \frac{2}{9} a^{3}$, $\frac{1}{9} a^{10} + \frac{2}{9} a^{4}$, $\frac{1}{9} a^{11} + \frac{2}{9} a^{5}$, $\frac{1}{45} a^{12} - \frac{2}{45} a^{9} + \frac{4}{9} a^{6} + \frac{14}{45} a^{3} - \frac{1}{5}$, $\frac{1}{135} a^{13} - \frac{1}{27} a^{11} + \frac{1}{45} a^{10} - \frac{1}{27} a^{9} + \frac{4}{27} a^{7} + \frac{1}{3} a^{6} + \frac{7}{27} a^{5} + \frac{8}{45} a^{4} + \frac{7}{27} a^{3} + \frac{1}{3} a^{2} - \frac{2}{5} a$, $\frac{1}{135} a^{14} + \frac{1}{135} a^{12} + \frac{1}{45} a^{11} - \frac{1}{27} a^{10} + \frac{1}{45} a^{9} + \frac{1}{27} a^{8} + \frac{4}{27} a^{6} + \frac{8}{45} a^{5} + \frac{7}{27} a^{4} + \frac{8}{45} a^{3} + \frac{17}{45} a^{2} + \frac{1}{3} a - \frac{2}{5}$, $\frac{1}{1646325} a^{15} - \frac{154}{36585} a^{12} - \frac{26069}{1646325} a^{9} + \frac{82121}{182925} a^{6} - \frac{399956}{1646325} a^{3} + \frac{11671}{60975}$, $\frac{1}{4938975} a^{16} - \frac{154}{109755} a^{13} + \frac{156856}{4938975} a^{10} + \frac{82121}{548775} a^{7} + \frac{1612219}{4938975} a^{4} + \frac{11671}{182925} a$, $\frac{1}{44450775} a^{17} - \frac{1}{14816925} a^{16} + \frac{1}{4938975} a^{15} - \frac{154}{987795} a^{14} + \frac{154}{329265} a^{13} + \frac{659}{109755} a^{12} - \frac{1489469}{44450775} a^{11} - \frac{705631}{14816925} a^{10} + \frac{266611}{4938975} a^{9} + \frac{82121}{4938975} a^{8} - \frac{82121}{1646325} a^{7} + \frac{163421}{548775} a^{6} + \frac{8197519}{44450775} a^{5} + \frac{7168181}{14816925} a^{4} - \frac{2448716}{4938975} a^{3} + \frac{743371}{1646325} a^{2} - \frac{11671}{548775} a - \frac{61499}{182925}$
Class group and class number
$C_{3}\times C_{6}\times C_{6}$, which has order $108$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 4441822.138935305 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_3:S_3$ (as 18T12):
| A solvable group of order 36 |
| The 12 conjugacy class representatives for $C_2\times C_3:S_3$ |
| Character table for $C_2\times C_3:S_3$ |
Intermediate fields
| \(\Q(\sqrt{-21}) \), 3.1.1323.1, 3.1.5292.1, 3.1.108.1, 3.1.588.1, 6.0.2352442176.1, 6.0.192036096.2, 6.0.9409768704.1, 6.0.116169984.1, 9.1.444611571264.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 18 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 | R | ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.8.3 | $x^{6} + 2 x^{3} + 6$ | $6$ | $1$ | $8$ | $D_{6}$ | $[2]_{3}^{2}$ |
| 2.12.16.13 | $x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$ | $6$ | $2$ | $16$ | $D_6$ | $[2]_{3}^{2}$ | |
| $3$ | 3.6.7.4 | $x^{6} + 3 x^{2} + 3$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ |
| 3.6.7.4 | $x^{6} + 3 x^{2} + 3$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ | |
| 3.6.7.4 | $x^{6} + 3 x^{2} + 3$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ | |
| 7 | Data not computed | ||||||