Normalized defining polynomial
\( x^{12} + 144x^{6} - 648x^{3} + 729 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
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)))
| |
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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{9}a^{6}$, $\frac{1}{9}a^{7}$, $\frac{1}{27}a^{8}+\frac{1}{3}a^{2}$, $\frac{1}{1215}a^{9}-\frac{2}{135}a^{6}+\frac{7}{135}a^{3}-\frac{7}{15}$, $\frac{1}{1215}a^{10}-\frac{2}{135}a^{7}+\frac{7}{135}a^{4}-\frac{7}{15}a$, $\frac{1}{3645}a^{11}-\frac{2}{405}a^{8}+\frac{52}{405}a^{5}-\frac{22}{45}a^{2}$
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)
| |
Torsion generator: | \( \frac{13}{1215} a^{9} + \frac{4}{135} a^{6} + \frac{226}{135} a^{3} - \frac{46}{15} \) (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)
| |
Fundamental units: | $\frac{34}{1215}a^{9}+\frac{7}{135}a^{6}+\frac{553}{135}a^{3}-\frac{163}{15}$, $\frac{61}{3645}a^{11}-\frac{4}{243}a^{10}-\frac{2}{405}a^{9}+\frac{13}{405}a^{8}-\frac{1}{27}a^{7}-\frac{1}{45}a^{6}+\frac{1012}{405}a^{5}-\frac{64}{27}a^{4}-\frac{29}{45}a^{3}-\frac{262}{45}a^{2}+\frac{16}{3}a-\frac{6}{5}$, $\frac{34}{1215}a^{11}+\frac{4}{405}a^{10}-\frac{73}{1215}a^{9}+\frac{7}{135}a^{8}+\frac{2}{45}a^{7}-\frac{19}{135}a^{6}+\frac{553}{135}a^{5}+\frac{73}{45}a^{4}-\frac{1186}{135}a^{3}-\frac{163}{15}a^{2}-\frac{3}{5}a+\frac{286}{15}$, $\frac{43}{1215}a^{11}+\frac{62}{1215}a^{10}+\frac{113}{1215}a^{9}+\frac{19}{135}a^{8}+\frac{26}{135}a^{7}+\frac{44}{135}a^{6}+\frac{751}{135}a^{5}+\frac{1064}{135}a^{4}+\frac{1916}{135}a^{3}-\frac{16}{15}a^{2}-\frac{44}{15}a-\frac{191}{15}$, $\frac{14}{243}a^{11}+\frac{1}{243}a^{10}+\frac{34}{1215}a^{9}+\frac{4}{27}a^{8}+\frac{1}{27}a^{7}+\frac{7}{135}a^{6}+\frac{233}{27}a^{5}+\frac{25}{27}a^{4}+\frac{553}{135}a^{3}-16a^{2}+\frac{11}{3}a-\frac{118}{15}$ | 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: | \( 24109.5003708 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 24109.5003708 \cdot 1}{8\cdot\sqrt{58498535041007616}}\cr\approx \mathstrut & 0.766663766499 \end{aligned}\]
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
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{8})\), 6.0.60466176.2 |
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: | 12.0.32905425960566784.34 |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
\(3\) | 3.6.10.6 | $x^{6} - 18 x^{4} - 12 x^{3} + 162 x^{2} + 432 x + 360$ | $3$ | $2$ | $10$ | $S_3^2$ | $[3/2, 5/2]_{2}^{2}$ |
3.6.10.6 | $x^{6} - 18 x^{4} - 12 x^{3} + 162 x^{2} + 432 x + 360$ | $3$ | $2$ | $10$ | $S_3^2$ | $[3/2, 5/2]_{2}^{2}$ |