Normalized defining polynomial
\( x^{18} - 6 x^{15} + 12 x^{14} + 15 x^{12} - 24 x^{11} + 36 x^{10} - 16 x^{9} - 18 x^{8} + 36 x^{7} - 15 x^{6} + 12 x^{5} + 6 x^{3} - 6 x^{2} + 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2426051245220667850752=2^{33}\cdot 3^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.42$ | 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: | $2, 3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{5}$, $\frac{1}{21} a^{15} - \frac{1}{21} a^{14} - \frac{1}{7} a^{13} - \frac{2}{21} a^{12} + \frac{1}{21} a^{11} + \frac{2}{21} a^{10} + \frac{1}{21} a^{9} + \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{1}{3} a^{6} - \frac{4}{21} a^{5} - \frac{2}{7} a^{4} + \frac{1}{3} a^{3} - \frac{8}{21} a^{2} + \frac{5}{21} a - \frac{2}{21}$, $\frac{1}{4683} a^{16} + \frac{106}{4683} a^{15} + \frac{157}{1561} a^{14} - \frac{159}{1561} a^{13} - \frac{113}{1561} a^{12} + \frac{174}{1561} a^{11} - \frac{80}{1561} a^{10} - \frac{664}{4683} a^{9} + \frac{237}{1561} a^{8} + \frac{1678}{4683} a^{7} + \frac{10}{4683} a^{6} + \frac{84}{223} a^{5} + \frac{423}{1561} a^{4} + \frac{723}{1561} a^{3} - \frac{678}{1561} a^{2} - \frac{9}{1561} a + \frac{563}{4683}$, $\frac{1}{79611} a^{17} - \frac{4}{79611} a^{16} + \frac{407}{79611} a^{15} + \frac{9484}{79611} a^{14} - \frac{9194}{79611} a^{13} - \frac{1891}{26537} a^{12} + \frac{1296}{26537} a^{11} - \frac{1024}{79611} a^{10} + \frac{8858}{79611} a^{9} - \frac{6956}{79611} a^{8} - \frac{35383}{79611} a^{7} - \frac{21190}{79611} a^{6} + \frac{299}{669} a^{5} + \frac{757}{11373} a^{4} - \frac{2158}{26537} a^{3} - \frac{580}{26537} a^{2} + \frac{11561}{79611} a + \frac{30392}{79611}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 5041.758308629595 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 36 |
| The 9 conjugacy class representatives for $S_3^2$ |
| Character table for $S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 3.1.216.1, 3.1.108.1, 6.2.1492992.5 x2, 6.2.373248.1, 6.2.1492992.4, 9.1.4353564672.1 |
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 6 sibling: | data not computed |
| Degree 9 sibling: | data not computed |
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | 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 | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.11.1 | $x^{6} + 14$ | $6$ | $1$ | $11$ | $D_{6}$ | $[3]_{3}^{2}$ |
| 2.12.22.60 | $x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$ | $6$ | $2$ | $22$ | $D_6$ | $[3]_{3}^{2}$ | |
| 3 | Data not computed | ||||||