Normalized defining polynomial
\( x^{12} - 8x^{9} + 12x^{7} + 20x^{6} - 54x^{4} + 12x^{3} + 24x^{2} - 6 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-106993205379072\) \(\medspace = -\,2^{26}\cdot 3^{13}\) | 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: | \(14.76\) | 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: | $2^{7/3}3^{7/6}\approx 18.157028993074817$ | ||
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: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{1289957}a^{11}-\frac{176630}{1289957}a^{10}+\frac{546855}{1289957}a^{9}-\frac{308455}{1289957}a^{8}-\frac{217202}{1289957}a^{7}-\frac{221865}{1289957}a^{6}+\frac{411267}{1289957}a^{5}+\frac{548288}{1289957}a^{4}-\frac{587719}{1289957}a^{3}-\frac{482593}{1289957}a^{2}+\frac{43054}{1289957}a-\frac{331505}{1289957}$
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)
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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: | $\frac{767623}{1289957}a^{11}+\frac{839823}{1289957}a^{10}+\frac{668725}{1289957}a^{9}-\frac{5545115}{1289957}a^{8}-\frac{6178467}{1289957}a^{7}+\frac{4345815}{1289957}a^{6}+\frac{21021258}{1289957}a^{5}+\frac{20978475}{1289957}a^{4}-\frac{24149854}{1289957}a^{3}-\frac{20274491}{1289957}a^{2}+\frac{5702130}{1289957}a+\frac{6694517}{1289957}$, $\frac{518814}{1289957}a^{11}+\frac{428460}{1289957}a^{10}+\frac{307476}{1289957}a^{9}-\frac{3866778}{1289957}a^{8}-\frac{3244693}{1289957}a^{7}+\frac{3934742}{1289957}a^{6}+\frac{13479495}{1289957}a^{5}+\frac{11072362}{1289957}a^{4}-\frac{20028832}{1289957}a^{3}-\frac{10830486}{1289957}a^{2}+\frac{5282372}{1289957}a+\frac{4401611}{1289957}$, $\frac{387496}{1289957}a^{11}+\frac{409983}{1289957}a^{10}+\frac{308776}{1289957}a^{9}-\frac{2822888}{1289957}a^{8}-\frac{2951684}{1289957}a^{7}+\frac{2544053}{1289957}a^{6}+\frac{10769394}{1289957}a^{5}+\frac{9938733}{1289957}a^{4}-\frac{13622715}{1289957}a^{3}-\frac{10690408}{1289957}a^{2}+\frac{4108774}{1289957}a+\frac{3506365}{1289957}$, $\frac{898736}{1289957}a^{11}+\frac{948654}{1289957}a^{10}+\frac{788409}{1289957}a^{9}-\frac{6563623}{1289957}a^{8}-\frac{7093561}{1289957}a^{7}+\frac{4780377}{1289957}a^{6}+\frac{24558786}{1289957}a^{5}+\frac{24519237}{1289957}a^{4}-\frac{27459663}{1289957}a^{3}-\frac{23389607}{1289957}a^{2}+\frac{4499443}{1289957}a+\frac{7180567}{1289957}$, $\frac{1454543}{1289957}a^{11}+\frac{935729}{1289957}a^{10}+\frac{507269}{1289957}a^{9}-\frac{11436551}{1289957}a^{8}-\frac{7569773}{1289957}a^{7}+\frac{13129436}{1289957}a^{6}+\frac{38285554}{1289957}a^{5}+\frac{24806059}{1289957}a^{4}-\frac{64811625}{1289957}a^{3}-\frac{27328277}{1289957}a^{2}+\frac{18411241}{1289957}a+\frac{13128869}{1289957}$, $\frac{797363}{1289957}a^{11}+\frac{568527}{1289957}a^{10}+\frac{358569}{1289957}a^{9}-\frac{6112588}{1289957}a^{8}-\frac{4371334}{1289957}a^{7}+\frac{6790684}{1289957}a^{6}+\frac{20729564}{1289957}a^{5}+\frac{14267373}{1289957}a^{4}-\frac{33025263}{1289957}a^{3}-\frac{14078944}{1289957}a^{2}+\frac{11650574}{1289957}a+\frac{5587211}{1289957}$ | 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: | \( 321.190145319 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 321.190145319 \cdot 1}{2\cdot\sqrt{106993205379072}}\cr\approx \mathstrut & 0.608153835419 \end{aligned}\]
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, 6.2.746496.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.2239488.1, some data not computed |
Degree 8 siblings: | 8.4.1719926784.2, 8.0.191102976.6 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.746496.2 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.26.8 | $x^{12} + 4 x^{8} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 2$ | $12$ | $1$ | $26$ | $C_2 \times S_4$ | $[2, 8/3, 8/3]_{3}^{2}$ |
\(3\) | 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 | $x^{6} + 3 x^{2} + 6$ | $6$ | $1$ | $7$ | $D_{6}$ | $[3/2]_{2}^{2}$ |