Normalized defining polynomial
\( x^{12} - 4x^{9} - 3x^{8} + 12x^{7} - 16x^{6} + 24x^{5} - 45x^{4} + 52x^{3} - 24x^{2} + 12x - 3 \)
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: | \(-427972821516288\) \(\medspace = -\,2^{28}\cdot 3^{13}\) | 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: | \(16.57\) | 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: | $2^{31/12}3^{7/6}\approx 21.592468065875266$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{9}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{10}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{4885}a^{11}+\frac{287}{4885}a^{10}+\frac{301}{4885}a^{9}+\frac{407}{4885}a^{8}+\frac{304}{977}a^{7}-\frac{1444}{4885}a^{6}-\frac{1173}{4885}a^{5}-\frac{539}{4885}a^{4}-\frac{2326}{4885}a^{3}+\frac{758}{4885}a^{2}-\frac{349}{4885}a+\frac{2434}{4885}$
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: | $\frac{214}{977}a^{11}+\frac{312}{4885}a^{10}+\frac{637}{4885}a^{9}-\frac{3183}{4885}a^{8}-\frac{3236}{4885}a^{7}+\frac{11281}{4885}a^{6}-\frac{16274}{4885}a^{5}+\frac{4825}{977}a^{4}-\frac{44366}{4885}a^{3}+\frac{43138}{4885}a^{2}-\frac{24641}{4885}a+\frac{7514}{4885}$, $\frac{3063}{4885}a^{11}+\frac{758}{4885}a^{10}-\frac{65}{977}a^{9}-\frac{12712}{4885}a^{8}-\frac{12346}{4885}a^{7}+\frac{35079}{4885}a^{6}-\frac{38573}{4885}a^{5}+\frac{58793}{4885}a^{4}-\frac{117494}{4885}a^{3}+\frac{127412}{4885}a^{2}-\frac{31413}{4885}a+\frac{17441}{4885}$, $\frac{1427}{4885}a^{11}+\frac{1163}{4885}a^{10}+\frac{1601}{4885}a^{9}-\frac{3457}{4885}a^{8}-\frac{5762}{4885}a^{7}+\frac{11629}{4885}a^{6}-\frac{16879}{4885}a^{5}+\frac{22217}{4885}a^{4}-\frac{52114}{4885}a^{3}+\frac{39207}{4885}a^{2}-\frac{3077}{977}a+\frac{8876}{4885}$, $\frac{973}{4885}a^{11}+\frac{806}{4885}a^{10}+\frac{150}{977}a^{9}-\frac{521}{977}a^{8}-\frac{4126}{4885}a^{7}+\frac{8707}{4885}a^{6}-\frac{9963}{4885}a^{5}+\frac{11926}{4885}a^{4}-\frac{29776}{4885}a^{3}+\frac{23347}{4885}a^{2}-\frac{5443}{4885}a-\frac{943}{4885}$, $\frac{55}{977}a^{11}+\frac{153}{977}a^{10}+\frac{707}{4885}a^{9}-\frac{86}{977}a^{8}-\frac{4064}{4885}a^{7}-\frac{3369}{4885}a^{6}-\frac{1142}{4885}a^{5}+\frac{1256}{4885}a^{4}+\frac{8101}{4885}a^{3}-\frac{4536}{4885}a^{2}+\frac{10518}{4885}a+\frac{1082}{4885}$, $\frac{2429}{4885}a^{11}+\frac{1499}{4885}a^{10}+\frac{262}{977}a^{9}-\frac{7937}{4885}a^{8}-\frac{2150}{977}a^{7}+\frac{4485}{977}a^{6}-\frac{27641}{4885}a^{5}+\frac{40006}{4885}a^{4}-\frac{86816}{4885}a^{3}+\frac{15344}{977}a^{2}-\frac{23133}{4885}a+\frac{15991}{4885}$ | 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: | \( 1199.61371106 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 1199.61371106 \cdot 1}{2\cdot\sqrt{427972821516288}}\cr\approx \mathstrut & 1.13569748331 \end{aligned}\]
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: | 6.2.4478976.3, 6.0.1492992.3 |
Degree 8 siblings: | 8.4.6879707136.1, 8.0.764411904.6 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.1492992.3 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.28.94 | $x^{12} + 6 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{4} + 2$ | $12$ | $1$ | $28$ | $C_2 \times S_4$ | $[8/3, 8/3, 3]_{3}^{2}$ |
\(3\) | 3.3.3.2 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
3.3.3.2 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
3.6.7.4 | $x^{6} + 3 x^{2} + 3$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ |