Normalized defining polynomial
\( x^{12} - 2x^{11} + 3x^{10} - 6x^{9} + 4x^{8} + 4x^{7} + x^{6} + 4x^{5} + 4x^{4} - 6x^{3} + 3x^{2} - 2x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(10497600000000\) \(\medspace = 2^{12}\cdot 3^{8}\cdot 5^{8}\) | 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: | \(12.16\) | 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: | $2^{7/6}3^{3/4}5^{2/3}\approx 14.963135389464592$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{15}a^{9}-\frac{1}{3}a^{7}-\frac{7}{15}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{7}{15}a^{3}-\frac{1}{3}a^{2}+\frac{1}{15}$, $\frac{1}{15}a^{10}+\frac{1}{5}a^{7}+\frac{1}{3}a^{5}+\frac{1}{5}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{4}{15}a+\frac{1}{3}$, $\frac{1}{45}a^{11}-\frac{1}{45}a^{10}-\frac{1}{45}a^{9}-\frac{7}{45}a^{8}-\frac{1}{15}a^{7}-\frac{8}{45}a^{6}-\frac{7}{45}a^{5}-\frac{2}{5}a^{4}+\frac{22}{45}a^{3}-\frac{14}{45}a^{2}+\frac{4}{45}a+\frac{14}{45}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -\frac{2}{5} a^{11} + \frac{4}{5} a^{10} - \frac{4}{3} a^{9} + \frac{37}{15} a^{8} - \frac{8}{5} a^{7} - \frac{4}{3} a^{6} + \frac{2}{15} a^{5} - \frac{44}{15} a^{4} - \frac{8}{3} a^{3} + \frac{8}{5} a^{2} - \frac{28}{15} a + \frac{4}{3} \) (order $6$) | 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: | $\frac{1}{3}a^{11}-\frac{4}{15}a^{10}+\frac{1}{5}a^{9}-\frac{2}{3}a^{8}-\frac{17}{15}a^{7}+\frac{44}{15}a^{6}+\frac{5}{3}a^{5}+\frac{6}{5}a^{4}+\frac{64}{15}a^{3}+\frac{2}{3}a^{2}-\frac{3}{5}a+\frac{6}{5}$, $\frac{31}{45}a^{11}-\frac{8}{9}a^{10}+\frac{53}{45}a^{9}-\frac{127}{45}a^{8}+\frac{199}{45}a^{6}+\frac{113}{45}a^{5}+\frac{13}{3}a^{4}+\frac{184}{45}a^{3}-\frac{44}{45}a^{2}-\frac{4}{9}a-\frac{22}{45}$, $\frac{17}{45}a^{11}-\frac{41}{45}a^{10}+\frac{73}{45}a^{9}-\frac{134}{45}a^{8}+\frac{13}{5}a^{7}+\frac{14}{45}a^{6}-\frac{14}{45}a^{5}+\frac{44}{15}a^{4}+\frac{89}{45}a^{3}-\frac{148}{45}a^{2}+\frac{134}{45}a-\frac{77}{45}$, $\frac{47}{45}a^{11}-\frac{77}{45}a^{10}+\frac{112}{45}a^{9}-\frac{239}{45}a^{8}+\frac{11}{5}a^{7}+\frac{221}{45}a^{6}+\frac{136}{45}a^{5}+\frac{73}{15}a^{4}+\frac{296}{45}a^{3}-\frac{178}{45}a^{2}+\frac{38}{45}a-\frac{98}{45}$, $\frac{2}{15}a^{11}-\frac{7}{15}a^{10}+\frac{1}{5}a^{9}-\frac{3}{5}a^{8}+\frac{3}{5}a^{7}+\frac{34}{15}a^{6}-\frac{14}{15}a^{5}-\frac{46}{15}a^{4}-\frac{41}{15}a^{3}-\frac{26}{5}a^{2}-\frac{52}{15}a-\frac{2}{15}$ | 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: | \( 176.860049974 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 176.860049974 \cdot 1}{6\cdot\sqrt{10497600000000}}\cr\approx \mathstrut & 0.559773961548 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.1.300.1 x3, 6.0.1080000.1, 6.2.3240000.1, 6.0.270000.1 |
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: | 4.2.10800.2 |
Degree 6 siblings: | 6.2.3240000.1, 6.0.1080000.1 |
Degree 8 sibling: | 8.0.116640000.2 |
Degree 12 sibling: | 12.2.125971200000000.1 |
Minimal sibling: | 4.2.10800.2 |
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 | R | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.28 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 130 x^{7} + 159 x^{6} + 132 x^{5} + 10 x^{4} - 100 x^{3} - 53 x^{2} + 22 x + 19$ | $6$ | $2$ | $12$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(5\) | 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |