Normalized defining polynomial
\( x^{12} - 3x^{8} - 6x^{7} - 2x^{6} - 6x^{5} - 3x^{4} + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-16325867520000\) \(\medspace = -\,2^{14}\cdot 3^{13}\cdot 5^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.62\) | 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: | $2^{7/6}3^{7/6}5^{1/2}\approx 18.08540037460562$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{8}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $a^{11}-3a^{7}-6a^{6}-2a^{5}-6a^{4}-3a^{3}$, $\frac{1}{3}a^{11}+\frac{2}{3}a^{10}+\frac{1}{3}a^{9}-\frac{2}{3}a^{8}-\frac{4}{3}a^{7}-\frac{11}{3}a^{6}-\frac{16}{3}a^{5}-\frac{11}{3}a^{4}-\frac{4}{3}a^{3}-\frac{4}{3}a^{2}+\frac{4}{3}a+\frac{5}{3}$, $\frac{1}{3}a^{11}+\frac{2}{3}a^{10}-\frac{1}{3}a^{9}-\frac{2}{3}a^{8}-\frac{4}{3}a^{7}-\frac{10}{3}a^{6}-\frac{10}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{5}{3}a^{2}+\frac{4}{3}a+\frac{4}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{8}-a^{7}-2a^{6}-\frac{4}{3}a^{5}-3a^{4}-2a^{3}-\frac{1}{3}a^{2}-a+1$, $\frac{2}{3}a^{11}+\frac{1}{3}a^{10}-\frac{2}{3}a^{9}-\frac{1}{3}a^{8}-\frac{5}{3}a^{7}-\frac{14}{3}a^{6}-\frac{5}{3}a^{5}-\frac{1}{3}a^{4}-\frac{4}{3}a^{3}+\frac{4}{3}a^{2}+\frac{8}{3}a+\frac{2}{3}$, $a^{11}+\frac{1}{3}a^{9}-3a^{7}-\frac{17}{3}a^{6}-3a^{5}-8a^{4}-\frac{10}{3}a^{3}-2a^{2}-2a+\frac{2}{3}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 115.889778794 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 115.889778794 \cdot 1}{2\cdot\sqrt{16325867520000}}\cr\approx \mathstrut & 0.561741277202 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T22):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2 \times S_4$ |
Character table for $C_2 \times S_4$ |
Intermediate fields
3.1.108.1, 6.2.1166400.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.699840.1, 6.2.233280.1 |
Degree 8 siblings: | 8.4.116640000.2, 8.0.1049760000.5 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.2.233280.1 |
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 | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.14.1 | $x^{12} + 2 x^{3} + 2$ | $12$ | $1$ | $14$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
\(3\) | 3.6.6.3 | $x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ |
3.6.7.2 | $x^{6} + 3 x^{2} + 6$ | $6$ | $1$ | $7$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
\(5\) | 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |