Normalized defining polynomial
\( x^{12} - x^{11} - 2x^{10} + 5x^{9} - x^{8} - 4x^{7} + 5x^{6} - 4x^{5} - x^{4} + 5x^{3} - 2x^{2} - x + 1 \)
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: | \(54507773961\) \(\medspace = 3^{6}\cdot 8647^{2}\) | 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: | \(7.85\) | 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: | $3^{1/2}8647^{1/2}\approx 161.06209982488122$ | ||
Ramified primes: | \(3\), \(8647\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{11}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{2}{5}$
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: | \( -\frac{8}{5} a^{11} + 2 a^{10} + 2 a^{9} - \frac{38}{5} a^{8} + \frac{21}{5} a^{7} + 2 a^{6} - \frac{32}{5} a^{5} + \frac{44}{5} a^{4} - 4 a^{3} - \frac{18}{5} a^{2} + \frac{14}{5} a - \frac{1}{5} \) (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: | $\frac{1}{5}a^{11}-\frac{9}{5}a^{10}+\frac{8}{5}a^{9}+3a^{8}-\frac{39}{5}a^{7}+\frac{17}{5}a^{6}+3a^{5}-\frac{36}{5}a^{4}+\frac{43}{5}a^{3}-3a^{2}-4a+\frac{13}{5}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{8}{5}a^{7}-\frac{8}{5}a^{6}-\frac{4}{5}a^{5}+\frac{12}{5}a^{4}-\frac{12}{5}a^{3}+\frac{4}{5}a^{2}+\frac{3}{5}a-\frac{9}{5}$, $\frac{4}{5}a^{11}-\frac{7}{5}a^{10}-\frac{1}{5}a^{9}+\frac{21}{5}a^{8}-\frac{24}{5}a^{7}+\frac{6}{5}a^{6}+\frac{19}{5}a^{5}-\frac{36}{5}a^{4}+\frac{29}{5}a^{3}-\frac{4}{5}a^{2}-\frac{18}{5}a+\frac{11}{5}$, $\frac{8}{5}a^{11}-\frac{9}{5}a^{10}-\frac{12}{5}a^{9}+\frac{37}{5}a^{8}-\frac{18}{5}a^{7}-\frac{13}{5}a^{6}+\frac{33}{5}a^{5}-\frac{47}{5}a^{4}+\frac{13}{5}a^{3}+\frac{22}{5}a^{2}-\frac{16}{5}a+\frac{7}{5}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{3}{5}a^{7}-\frac{3}{5}a^{6}+\frac{1}{5}a^{5}-\frac{3}{5}a^{4}-\frac{7}{5}a^{3}+\frac{4}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$ | 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: | \( 4.14786100391 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 4.14786100391 \cdot 1}{6\cdot\sqrt{54507773961}}\cr\approx \mathstrut & 0.182189326862 \end{aligned}\]
Galois group
$C_2\times S_6$ (as 12T219):
A non-solvable group of order 1440 |
The 22 conjugacy class representatives for $S_6\times C_2$ |
Character table for $S_6\times C_2$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 6.2.233469.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
\(8647\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |