Normalized defining polynomial
\( x^{12} - 2x^{8} - x^{7} + 2x^{6} + 4x^{4} - 2x^{3} - x^{2} - x + 1 \)
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: | \(567013518009\) \(\medspace = 3^{6}\cdot 167^{4}\) | 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: | \(9.54\) | 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: | $3^{1/2}167^{2/3}\approx 52.52567088234156$ | ||
Ramified primes: | \(3\), \(167\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{33}a^{11}-\frac{5}{33}a^{10}+\frac{1}{11}a^{9}-\frac{4}{33}a^{8}-\frac{5}{11}a^{7}-\frac{1}{11}a^{6}-\frac{16}{33}a^{5}+\frac{14}{33}a^{4}-\frac{1}{3}a^{3}+\frac{3}{11}a^{2}-\frac{2}{33}a+\frac{3}{11}$
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)
| |
Torsion generator: | \( -\frac{31}{33} a^{11} - \frac{32}{33} a^{10} - \frac{16}{33} a^{9} - \frac{8}{33} a^{8} + \frac{58}{33} a^{7} + \frac{31}{11} a^{6} + \frac{1}{33} a^{5} - \frac{16}{33} a^{4} - 4 a^{3} - \frac{70}{33} a^{2} + \frac{29}{33} a + \frac{62}{33} \) (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: | $\frac{32}{33}a^{11}+\frac{16}{33}a^{10}+\frac{8}{33}a^{9}+\frac{4}{33}a^{8}-\frac{62}{33}a^{7}-\frac{21}{11}a^{6}+\frac{16}{33}a^{5}+\frac{8}{33}a^{4}+4a^{3}+\frac{2}{33}a^{2}-\frac{31}{33}a-\frac{31}{33}$, $\frac{4}{11}a^{11}-\frac{5}{33}a^{10}-\frac{8}{33}a^{9}-\frac{5}{11}a^{8}-\frac{37}{33}a^{7}-\frac{1}{11}a^{6}+\frac{13}{11}a^{5}+\frac{14}{33}a^{4}+\frac{7}{3}a^{3}-\frac{35}{33}a^{2}-\frac{8}{11}a-\frac{35}{33}$, $\frac{41}{33}a^{11}+\frac{26}{33}a^{10}+\frac{8}{11}a^{9}+\frac{1}{33}a^{8}-\frac{29}{11}a^{7}-\frac{30}{11}a^{6}+\frac{4}{33}a^{5}+\frac{13}{33}a^{4}+\frac{19}{3}a^{3}+\frac{13}{11}a^{2}-\frac{16}{33}a-\frac{31}{11}$, $\frac{28}{33}a^{11}+\frac{1}{11}a^{10}-\frac{4}{33}a^{9}-\frac{13}{33}a^{8}-\frac{68}{33}a^{7}-\frac{17}{11}a^{6}+\frac{47}{33}a^{5}+\frac{6}{11}a^{4}+\frac{13}{3}a^{3}-\frac{34}{33}a^{2}-\frac{23}{33}a-\frac{67}{33}$, $\frac{52}{33}a^{11}+\frac{37}{33}a^{10}+\frac{8}{11}a^{9}+\frac{23}{33}a^{8}-\frac{29}{11}a^{7}-\frac{41}{11}a^{6}+\frac{26}{33}a^{5}+\frac{2}{33}a^{4}+\frac{17}{3}a^{3}+\frac{13}{11}a^{2}-\frac{38}{33}a-\frac{20}{11}$ | 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: | \( 16.9693695402 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 16.9693695402 \cdot 1}{6\cdot\sqrt{567013518009}}\cr\approx \mathstrut & 0.231098411705 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_6$ |
Character table for $S_6$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 6.0.83667.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.0.83667.1, 6.4.21000500667.1 |
Degree 10 sibling: | 10.4.585682963101963.1 |
Degree 12 sibling: | deg 12 |
Degree 15 siblings: | deg 15, deg 15 |
Degree 20 siblings: | deg 20, deg 20, deg 20 |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.0.83667.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(167\) | 167.2.0.1 | $x^{2} + 166 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
167.2.0.1 | $x^{2} + 166 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
167.2.0.1 | $x^{2} + 166 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
167.6.4.1 | $x^{6} + 498 x^{5} + 82683 x^{4} + 4579610 x^{3} + 496581 x^{2} + 13812996 x + 763519616$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |