Normalized defining polynomial
\( x^{12} - 12x^{9} + 26x^{6} + 60x^{3} - 2 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(320979616137216\) \(\medspace = 2^{26}\cdot 3^{14}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.18\) | 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))
| |
Galois root discriminant: | $2^{13/6}3^{25/18}\approx 20.649049558832502$ | ||
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)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{2}$, $\frac{1}{39}a^{9}+\frac{4}{39}a^{6}-\frac{14}{39}a^{3}-\frac{8}{39}$, $\frac{1}{117}a^{10}-\frac{1}{9}a^{8}-\frac{1}{13}a^{7}+\frac{1}{9}a^{6}-\frac{1}{3}a^{5}-\frac{14}{117}a^{4}+\frac{1}{3}a^{3}-\frac{4}{9}a^{2}-\frac{7}{39}a+\frac{4}{9}$, $\frac{1}{117}a^{11}-\frac{1}{117}a^{9}-\frac{1}{13}a^{8}+\frac{1}{9}a^{7}+\frac{1}{13}a^{6}-\frac{14}{117}a^{5}+\frac{1}{3}a^{4}+\frac{14}{117}a^{3}-\frac{7}{39}a^{2}+\frac{4}{9}a+\frac{7}{39}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{1}{39}a^{9}-\frac{3}{13}a^{6}+\frac{25}{39}a^{3}-\frac{7}{13}$, $\frac{2}{39}a^{11}-\frac{5}{117}a^{10}-\frac{67}{117}a^{8}+\frac{5}{13}a^{7}+\frac{1}{9}a^{6}+\frac{37}{39}a^{5}-\frac{47}{117}a^{4}-\frac{2}{3}a^{3}+\frac{407}{117}a^{2}-\frac{43}{39}a-\frac{5}{9}$, $\frac{2}{9}a^{11}+\frac{2}{39}a^{10}+\frac{4}{117}a^{9}-\frac{8}{3}a^{8}-\frac{67}{117}a^{7}-\frac{4}{13}a^{6}+\frac{53}{9}a^{5}+\frac{37}{39}a^{4}+\frac{61}{117}a^{3}+\frac{38}{3}a^{2}+\frac{407}{117}a+\frac{11}{39}$, $\frac{2}{39}a^{11}+\frac{5}{117}a^{10}-\frac{80}{117}a^{8}-\frac{5}{13}a^{7}-\frac{1}{9}a^{6}+\frac{21}{13}a^{5}+\frac{47}{117}a^{4}+\frac{2}{3}a^{3}+\frac{472}{117}a^{2}+\frac{82}{39}a+\frac{5}{9}$, $\frac{1}{13}a^{9}-\frac{9}{13}a^{6}+\frac{12}{13}a^{3}+\frac{5}{13}$, $\frac{2}{9}a^{11}+\frac{2}{39}a^{10}+\frac{1}{117}a^{9}-\frac{8}{3}a^{8}-\frac{67}{117}a^{7}-\frac{1}{13}a^{6}+\frac{53}{9}a^{5}+\frac{37}{39}a^{4}-\frac{14}{117}a^{3}+\frac{38}{3}a^{2}+\frac{407}{117}a+\frac{71}{39}$, $\frac{1}{117}a^{11}-\frac{2}{39}a^{10}-\frac{4}{117}a^{9}-\frac{1}{13}a^{8}+\frac{67}{117}a^{7}+\frac{4}{13}a^{6}-\frac{14}{117}a^{5}-\frac{37}{39}a^{4}-\frac{61}{117}a^{3}+\frac{71}{39}a^{2}-\frac{407}{117}a-\frac{11}{39}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 602.053815744 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 602.053815744 \cdot 1}{2\cdot\sqrt{320979616137216}}\cr\approx \mathstrut & 0.418992304992 \end{aligned}\]
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
\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 6.2.1492992.5 |
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: | 12.0.35664401793024.5 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.26.45 | $x^{12} + 4 x^{11} + 2 x^{10} + 6 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 14$ | $12$ | $1$ | $26$ | $S_3 \times C_2^2$ | $[2, 3]_{3}^{2}$ |
\(3\) | 3.12.14.5 | $x^{12} + 6 x^{6} + 36 x^{4} + 9$ | $6$ | $2$ | $14$ | $S_3^2$ | $[3/2, 3/2]_{2}^{2}$ |