Normalized defining polynomial
\( x^{12} - 6x^{10} - 12x^{9} - 12x^{8} + 14x^{6} + 12x^{5} - 9x^{4} - 8x^{3} + 12x^{2} - 4 \)
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: | \(-855945643032576\) \(\medspace = -\,2^{29}\cdot 3^{13}\) | 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: | \(17.55\) | 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: | $2^{11/4}3^{7/6}\approx 24.23672593327708$ | ||
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: | \(\Q(\sqrt{-6}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{20}a^{10}+\frac{1}{20}a^{9}+\frac{1}{20}a^{8}-\frac{1}{4}a^{7}-\frac{1}{20}a^{6}-\frac{1}{20}a^{5}-\frac{3}{20}a^{4}+\frac{3}{20}a^{3}+\frac{3}{10}a^{2}+\frac{3}{10}a+\frac{1}{5}$, $\frac{1}{200}a^{11}-\frac{1}{100}a^{10}-\frac{1}{100}a^{9}-\frac{1}{25}a^{8}-\frac{23}{100}a^{7}-\frac{1}{25}a^{6}-\frac{1}{10}a^{5}+\frac{13}{50}a^{4}-\frac{63}{200}a^{3}+\frac{9}{100}a^{2}-\frac{37}{100}a-\frac{13}{50}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{43}{200}a^{11}+\frac{17}{100}a^{10}-\frac{133}{100}a^{9}-\frac{337}{100}a^{8}-\frac{439}{100}a^{7}-\frac{257}{100}a^{6}+\frac{3}{5}a^{5}+\frac{63}{100}a^{4}-\frac{849}{200}a^{3}-\frac{239}{50}a^{2}+\frac{269}{100}a+\frac{68}{25}$, $\frac{13}{25}a^{11}-\frac{11}{25}a^{10}-\frac{61}{25}a^{9}-\frac{114}{25}a^{8}-\frac{98}{25}a^{7}+\frac{31}{25}a^{6}+6a^{5}+\frac{131}{25}a^{4}-\frac{74}{25}a^{3}+\frac{24}{25}a^{2}+\frac{3}{25}a-\frac{41}{25}$, $\frac{101}{200}a^{11}+\frac{29}{100}a^{10}-\frac{271}{100}a^{9}-\frac{799}{100}a^{8}-\frac{1123}{100}a^{7}-\frac{659}{100}a^{6}+\frac{23}{5}a^{5}+\frac{1161}{100}a^{4}+\frac{717}{200}a^{3}-\frac{109}{25}a^{2}-\frac{157}{100}a+\frac{11}{25}$, $\frac{11}{40}a^{11}-\frac{1}{5}a^{10}-\frac{6}{5}a^{9}-\frac{13}{5}a^{8}-\frac{29}{10}a^{7}-\frac{4}{5}a^{6}+\frac{23}{20}a^{5}+2a^{4}-\frac{71}{40}a^{3}+\frac{4}{5}a^{2}+\frac{3}{4}a-\frac{2}{5}$, $\frac{13}{200}a^{11}-\frac{13}{100}a^{10}-\frac{19}{50}a^{9}-\frac{1}{50}a^{8}+\frac{19}{25}a^{7}+\frac{37}{25}a^{6}+\frac{19}{20}a^{5}-\frac{28}{25}a^{4}-\frac{469}{200}a^{3}+\frac{117}{100}a^{2}+\frac{169}{100}a-\frac{69}{50}$, $\frac{7}{40}a^{11}-\frac{1}{10}a^{10}-\frac{17}{20}a^{9}-\frac{33}{20}a^{8}-\frac{41}{20}a^{7}-\frac{3}{20}a^{6}+a^{5}+\frac{37}{20}a^{4}-\frac{1}{40}a^{3}+\frac{3}{20}a^{2}+\frac{41}{20}a-\frac{21}{10}$ | 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: | \( 3094.46806616 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 3094.46806616 \cdot 1}{2\cdot\sqrt{855945643032576}}\cr\approx \mathstrut & 2.07153487853 \end{aligned}\]
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
3.1.216.1, 6.2.2985984.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.17915904.2, some data not computed |
Degree 8 siblings: | 8.0.191102976.7, 8.4.27518828544.8 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.0.746496.1 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.9.4 | $x^{4} + 10 x^{2} + 8 x + 2$ | $4$ | $1$ | $9$ | $D_{4}$ | $[2, 3, 7/2]$ |
2.4.10.4 | $x^{4} + 4 x^{3} + 8 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ | |
2.4.10.4 | $x^{4} + 4 x^{3} + 8 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ | |
\(3\) | 3.6.6.4 | $x^{6} + 48 x^{4} + 6 x^{3} + 36 x^{2} + 36 x + 9$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ |
3.6.7.5 | $x^{6} + 6 x^{2} + 3$ | $6$ | $1$ | $7$ | $D_{6}$ | $[3/2]_{2}^{2}$ |