Normalized defining polynomial
\( x^{18} - 4x^{15} + 6x^{12} - 5x^{9} + 6x^{6} - 4x^{3} + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-6222062031041447163\) \(\medspace = -\,3^{21}\cdot 29^{6}\) | 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: | \(11.07\) | 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: | $3^{31/18}29^{1/2}\approx 35.71970195840605$ | ||
Ramified primes: | \(3\), \(29\) | 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: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{12}-\frac{1}{3}a^{9}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{15}+\frac{4}{9}a^{13}+\frac{4}{9}a^{12}+\frac{2}{9}a^{10}+\frac{2}{9}a^{9}+\frac{2}{9}a^{7}+\frac{2}{9}a^{6}+\frac{4}{9}a^{4}+\frac{4}{9}a^{3}+\frac{1}{9}a+\frac{1}{9}$, $\frac{1}{9}a^{17}-\frac{1}{9}a^{15}+\frac{1}{9}a^{14}+\frac{1}{3}a^{13}+\frac{2}{9}a^{12}+\frac{2}{9}a^{11}-\frac{2}{9}a^{9}-\frac{4}{9}a^{8}-\frac{1}{3}a^{7}+\frac{1}{9}a^{6}+\frac{4}{9}a^{5}-\frac{4}{9}a^{3}-\frac{2}{9}a^{2}+\frac{1}{3}a-\frac{4}{9}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{4}{3} a^{15} - \frac{14}{3} a^{12} + \frac{17}{3} a^{9} - \frac{10}{3} a^{6} + \frac{16}{3} a^{3} - \frac{5}{3} \) (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: | $\frac{4}{3}a^{16}-\frac{14}{3}a^{13}+\frac{17}{3}a^{10}-\frac{10}{3}a^{7}+\frac{16}{3}a^{4}-\frac{8}{3}a$, $\frac{1}{3}a^{17}-\frac{2}{3}a^{14}-\frac{1}{3}a^{11}+\frac{2}{3}a^{8}+\frac{4}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{4}{9}a^{16}+\frac{4}{9}a^{15}-\frac{11}{9}a^{13}-\frac{11}{9}a^{12}+\frac{8}{9}a^{10}+\frac{8}{9}a^{9}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{7}{9}a^{4}+\frac{7}{9}a^{3}+\frac{4}{9}a+\frac{4}{9}$, $\frac{4}{9}a^{16}-\frac{11}{9}a^{15}-\frac{11}{9}a^{13}+\frac{37}{9}a^{12}+\frac{8}{9}a^{10}-\frac{40}{9}a^{9}-\frac{1}{9}a^{7}+\frac{23}{9}a^{6}+\frac{7}{9}a^{4}-\frac{44}{9}a^{3}+\frac{4}{9}a+\frac{16}{9}$, $\frac{4}{9}a^{17}+\frac{5}{9}a^{16}+\frac{4}{9}a^{15}-\frac{17}{9}a^{14}-\frac{19}{9}a^{13}-\frac{17}{9}a^{12}+\frac{26}{9}a^{11}+\frac{28}{9}a^{10}+\frac{26}{9}a^{9}-\frac{22}{9}a^{8}-\frac{23}{9}a^{7}-\frac{22}{9}a^{6}+\frac{25}{9}a^{5}+\frac{29}{9}a^{4}+\frac{25}{9}a^{3}-\frac{20}{9}a^{2}-\frac{16}{9}a-\frac{11}{9}$, $\frac{8}{9}a^{17}-\frac{4}{3}a^{16}-\frac{5}{9}a^{15}-\frac{28}{9}a^{14}+\frac{13}{3}a^{13}+\frac{19}{9}a^{12}+\frac{34}{9}a^{11}-\frac{14}{3}a^{10}-\frac{28}{9}a^{9}-\frac{23}{9}a^{8}+\frac{8}{3}a^{7}+\frac{23}{9}a^{6}+\frac{32}{9}a^{5}-\frac{16}{3}a^{4}-\frac{29}{9}a^{3}-\frac{7}{9}a^{2}+\frac{4}{3}a+\frac{16}{9}$, $\frac{11}{9}a^{17}+\frac{7}{9}a^{15}-\frac{37}{9}a^{14}-\frac{26}{9}a^{12}+\frac{40}{9}a^{11}+\frac{32}{9}a^{9}-\frac{23}{9}a^{8}-\frac{22}{9}a^{6}+\frac{44}{9}a^{5}+\frac{37}{9}a^{3}-\frac{7}{9}a^{2}-\frac{20}{9}$, $\frac{4}{9}a^{16}-\frac{8}{9}a^{15}-\frac{1}{3}a^{14}-\frac{17}{9}a^{13}+\frac{28}{9}a^{12}+a^{11}+\frac{26}{9}a^{10}-\frac{34}{9}a^{9}-\frac{2}{3}a^{8}-\frac{22}{9}a^{7}+\frac{23}{9}a^{6}+\frac{25}{9}a^{4}-\frac{32}{9}a^{3}-\frac{4}{3}a^{2}-\frac{20}{9}a+\frac{7}{9}$ | 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: | \( 184.401892895 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 184.401892895 \cdot 1}{6\cdot\sqrt{6222062031041447163}}\cr\approx \mathstrut & 0.188046723779 \end{aligned}\]
Galois group
$C_3^3:D_6$ (as 18T119):
A solvable group of order 324 |
The 44 conjugacy class representatives for $C_3^3:D_6$ |
Character table for $C_3^3:D_6$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.1.87.1, 6.0.22707.1, 9.3.480048687.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 siblings: | data not computed |
Degree 36 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 | $18$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{6}$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ | $18$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.12.18.86 | $x^{12} + 12 x^{10} + 36 x^{8} + 33 x^{6} + 144 x^{4} + 54 x^{3} + 225$ | $6$ | $2$ | $18$ | $C_6\times S_3$ | $[3/2, 2]_{2}^{2}$ | |
\(29\) | 29.6.0.1 | $x^{6} + x^{4} + 25 x^{3} + 17 x^{2} + 13 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
29.12.6.1 | $x^{12} + 4176 x^{11} + 7266416 x^{10} + 6743683250 x^{9} + 3520736141840 x^{8} + 980530396296652 x^{7} + 113880628956515675 x^{6} + 29503543251836820 x^{5} + 141318186546148833 x^{4} + 2879679256127599138 x^{3} + 2302312844056605504 x^{2} + 1655554410097277308 x + 292378155115321601$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |