Normalized defining polynomial
\( x^{12} - 12x^{9} + 72x^{6} - 108x^{3} + 81 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(58498535041007616\) \(\medspace = 2^{24}\cdot 3^{20}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(24.96\) | 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))
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Galois root discriminant: | $2^{2}3^{37/18}\approx 38.265662534797954$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{9}a^{6}$, $\frac{1}{9}a^{7}$, $\frac{1}{9}a^{8}$, $\frac{1}{81}a^{9}+\frac{1}{27}a^{6}+\frac{1}{9}a^{3}+\frac{1}{3}$, $\frac{1}{81}a^{10}+\frac{1}{27}a^{7}+\frac{1}{9}a^{4}+\frac{1}{3}a$, $\frac{1}{81}a^{11}+\frac{1}{27}a^{8}+\frac{1}{9}a^{5}+\frac{1}{3}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{1}{81} a^{9} + \frac{5}{27} a^{6} - \frac{10}{9} a^{3} + \frac{5}{3} \) (order $8$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{1}{81}a^{9}-\frac{2}{27}a^{6}+\frac{4}{9}a^{3}-\frac{2}{3}$, $\frac{1}{81}a^{11}-\frac{2}{81}a^{10}+\frac{5}{81}a^{9}-\frac{5}{27}a^{8}+\frac{7}{27}a^{7}-\frac{19}{27}a^{6}+\frac{10}{9}a^{5}-\frac{14}{9}a^{4}+\frac{35}{9}a^{3}-\frac{8}{3}a^{2}+\frac{7}{3}a-\frac{10}{3}$, $\frac{1}{81}a^{11}-\frac{1}{27}a^{10}+\frac{4}{81}a^{9}-\frac{5}{27}a^{8}+\frac{4}{9}a^{7}-\frac{14}{27}a^{6}+\frac{10}{9}a^{5}-\frac{7}{3}a^{4}+\frac{25}{9}a^{3}-\frac{8}{3}a^{2}+2a-\frac{2}{3}$, $\frac{1}{27}a^{11}+\frac{1}{27}a^{10}+\frac{7}{81}a^{9}-\frac{1}{3}a^{8}-\frac{5}{9}a^{7}-\frac{26}{27}a^{6}+2a^{5}+\frac{10}{3}a^{4}+\frac{34}{9}a^{3}-2a-\frac{8}{3}$, $\frac{5}{81}a^{11}-\frac{4}{81}a^{10}-\frac{1}{81}a^{9}-\frac{19}{27}a^{8}+\frac{14}{27}a^{7}+\frac{2}{27}a^{6}+\frac{35}{9}a^{5}-\frac{22}{9}a^{4}-\frac{1}{9}a^{3}-\frac{10}{3}a^{2}+\frac{2}{3}a+\frac{8}{3}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 20443.6078668 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 20443.6078668 \cdot 1}{8\cdot\sqrt{58498535041007616}}\cr\approx \mathstrut & 0.650091174307 \end{aligned}\]
Galois group
$S_3\times D_6$ (as 12T37):
A solvable group of order 72 |
The 18 conjugacy class representatives for $S_3\times D_6$ |
Character table for $S_3\times D_6$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{8})\), 6.2.30233088.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.2 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
\(3\) | 3.6.10.4 | $x^{6} + 6 x^{3} + 81 x^{2} + 9$ | $3$ | $2$ | $10$ | $S_3^2$ | $[3/2, 5/2]_{2}^{2}$ |
3.6.10.4 | $x^{6} + 6 x^{3} + 81 x^{2} + 9$ | $3$ | $2$ | $10$ | $S_3^2$ | $[3/2, 5/2]_{2}^{2}$ |