Normalized defining polynomial
\( x^{12} - 9x^{8} - 30x^{6} - 18x^{5} - 45x^{4} - 22x^{3} - 36x^{2} - 3 \)
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: | \(-171382426877952\) \(\medspace = -\,2^{14}\cdot 3^{21}\) | 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: | \(15.35\) | 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: | $2^{7/6}3^{11/6}\approx 16.82379477331321$ | ||
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: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{6}+\frac{1}{3}a^{3}$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{3}a^{8}+\frac{4}{9}a^{7}+\frac{2}{9}a^{6}-\frac{1}{3}a^{5}+\frac{4}{9}a^{4}-\frac{1}{9}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{4649103}a^{11}-\frac{92102}{4649103}a^{10}-\frac{284870}{4649103}a^{9}+\frac{658810}{4649103}a^{8}-\frac{2275376}{4649103}a^{7}+\frac{614122}{4649103}a^{6}+\frac{2222026}{4649103}a^{5}+\frac{475390}{4649103}a^{4}-\frac{2217173}{4649103}a^{3}+\frac{655618}{1549701}a^{2}+\frac{370417}{1549701}a+\frac{520981}{1549701}$
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)
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Fundamental units: | $\frac{433973}{4649103}a^{11}+\frac{106946}{4649103}a^{10}-\frac{40936}{4649103}a^{9}-\frac{135061}{4649103}a^{8}-\frac{3967462}{4649103}a^{7}-\frac{562171}{4649103}a^{6}-\frac{13005859}{4649103}a^{5}-\frac{9918763}{4649103}a^{4}-\frac{17960851}{4649103}a^{3}-\frac{4593286}{1549701}a^{2}-\frac{4157092}{1549701}a-\frac{390181}{1549701}$, $\frac{351721}{4649103}a^{11}+\frac{225595}{4649103}a^{10}+\frac{123751}{4649103}a^{9}-\frac{180314}{4649103}a^{8}-\frac{2997944}{4649103}a^{7}-\frac{1853954}{4649103}a^{6}-\frac{12001574}{4649103}a^{5}-\frac{11707679}{4649103}a^{4}-\frac{25543172}{4649103}a^{3}-\frac{5539325}{1549701}a^{2}-\frac{6640784}{1549701}a-\frac{2370176}{1549701}$, $\frac{681743}{4649103}a^{11}-\frac{141802}{4649103}a^{10}-\frac{115657}{4649103}a^{9}+\frac{112907}{4649103}a^{8}-\frac{6085924}{4649103}a^{7}+\frac{986816}{4649103}a^{6}-\frac{19249006}{4649103}a^{5}-\frac{8596102}{4649103}a^{4}-\frac{24772045}{4649103}a^{3}-\frac{3329947}{1549701}a^{2}-\frac{6462160}{1549701}a+\frac{1760528}{1549701}$, $\frac{573547}{4649103}a^{11}+\frac{348760}{4649103}a^{10}-\frac{324326}{4649103}a^{9}-\frac{446657}{4649103}a^{8}-\frac{5005088}{4649103}a^{7}-\frac{2676422}{4649103}a^{6}-\frac{14510663}{4649103}a^{5}-\frac{17064857}{4649103}a^{4}-\frac{23038133}{4649103}a^{3}-\frac{4111202}{1549701}a^{2}-\frac{2633228}{1549701}a-\frac{574976}{1549701}$, $\frac{688604}{4649103}a^{11}+\frac{1257518}{4649103}a^{10}+\frac{80801}{4649103}a^{9}-\frac{269500}{4649103}a^{8}-\frac{6269353}{4649103}a^{7}-\frac{11040202}{4649103}a^{6}-\frac{21436759}{4649103}a^{5}-\frac{48325009}{4649103}a^{4}-\frac{55408435}{4649103}a^{3}-\frac{22513264}{1549701}a^{2}-\frac{16355536}{1549701}a-\frac{12379780}{1549701}$, $\frac{22451}{1549701}a^{11}-\frac{136490}{1549701}a^{10}+\frac{19094}{172189}a^{9}+\frac{80399}{1549701}a^{8}-\frac{295001}{1549701}a^{7}+\frac{389516}{516567}a^{6}-\frac{2302000}{1549701}a^{5}+\frac{3117316}{1549701}a^{4}-\frac{910805}{516567}a^{3}+\frac{219620}{516567}a^{2}-\frac{1185389}{516567}a-\frac{142176}{172189}$ | 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: | \( 721.332827695 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 721.332827695 \cdot 1}{2\cdot\sqrt{171382426877952}}\cr\approx \mathstrut & 1.07915018752 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
3.1.972.2, 4.2.3888.1 x2, 6.2.3779136.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 4 sibling: | 4.2.3888.1 |
Degree 6 siblings: | 6.2.3779136.3, 6.0.11337408.4 |
Degree 8 sibling: | 8.0.136048896.1 |
Degree 12 sibling: | 12.0.128536820158464.9 |
Minimal sibling: | 4.2.3888.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.2.0.1}{2} }^{6}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.14.1 | $x^{12} + 2 x^{3} + 2$ | $12$ | $1$ | $14$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
\(3\) | 3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ |
3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
3.6.11.7 | $x^{6} + 9 x^{2} + 12$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ |