Normalized defining polynomial
\( x^{18} - x^{15} + 2x^{12} - 9x^{9} + 12x^{6} - 5x^{3} + 1 \)
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: | \(-5040742784941043712\) \(\medspace = -\,2^{12}\cdot 3^{21}\cdot 7^{6}\) | 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: | \(10.94\) | 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: | $2^{2/3}3^{7/6}7^{2/3}\approx 20.927956356407396$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | 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: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $9$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{8}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{8}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{14}-\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{11}+\frac{1}{9}a^{8}-\frac{4}{9}a^{7}-\frac{1}{9}a^{6}-\frac{4}{9}a^{5}-\frac{2}{9}a^{4}-\frac{1}{9}a^{3}+\frac{1}{3}a^{2}+\frac{4}{9}a+\frac{1}{9}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{14}-\frac{1}{9}a^{11}+\frac{1}{9}a^{9}+\frac{4}{9}a^{7}+\frac{4}{9}a^{6}-\frac{1}{3}a^{4}+\frac{2}{9}a^{3}+\frac{4}{9}a^{2}-\frac{4}{9}a+\frac{1}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{14}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{3}a^{8}-\frac{1}{9}a^{7}+\frac{4}{9}a^{6}+\frac{1}{9}a^{5}+\frac{4}{9}a^{4}+\frac{2}{9}a^{3}+\frac{2}{9}a^{2}-\frac{1}{3}a-\frac{4}{9}$
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: | \( -\frac{2}{3} a^{15} + \frac{1}{3} a^{12} - \frac{4}{3} a^{9} + \frac{16}{3} a^{6} - 6 a^{3} + 2 \) (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: | $\frac{1}{3}a^{11}+\frac{1}{3}a^{8}+\frac{2}{3}a^{5}-\frac{5}{3}a^{2}$, $\frac{2}{3}a^{16}-\frac{1}{3}a^{13}+\frac{4}{3}a^{10}-\frac{16}{3}a^{7}+6a^{4}-2a$, $\frac{2}{9}a^{17}-\frac{2}{9}a^{16}-\frac{1}{9}a^{15}-\frac{2}{9}a^{14}+\frac{1}{9}a^{12}+\frac{5}{9}a^{11}-\frac{4}{9}a^{10}-\frac{2}{9}a^{9}-\frac{16}{9}a^{8}+\frac{4}{3}a^{7}+\frac{10}{9}a^{6}+3a^{5}-\frac{11}{9}a^{4}-\frac{8}{9}a^{3}-\frac{13}{9}a^{2}-\frac{5}{9}a+\frac{7}{9}$, $\frac{2}{9}a^{16}-\frac{2}{9}a^{15}+\frac{4}{9}a^{14}-\frac{4}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{3}a^{10}-\frac{4}{9}a^{9}+\frac{7}{9}a^{8}-\frac{23}{9}a^{7}+\frac{16}{9}a^{6}-\frac{28}{9}a^{5}+\frac{37}{9}a^{4}-2a^{3}+\frac{20}{9}a^{2}-\frac{16}{9}a$, $\frac{7}{9}a^{17}+\frac{2}{9}a^{16}-\frac{1}{9}a^{15}-\frac{5}{9}a^{14}-\frac{1}{9}a^{13}+\frac{1}{3}a^{12}+\frac{4}{3}a^{11}+\frac{4}{9}a^{10}-\frac{1}{9}a^{9}-\frac{61}{9}a^{8}-\frac{16}{9}a^{7}+\frac{13}{9}a^{6}+\frac{65}{9}a^{5}+2a^{4}-\frac{20}{9}a^{3}-\frac{14}{9}a^{2}+1$, $\frac{2}{9}a^{17}-\frac{2}{9}a^{15}-\frac{5}{9}a^{14}+\frac{1}{9}a^{13}+\frac{1}{3}a^{12}+\frac{4}{9}a^{11}+\frac{2}{9}a^{10}-\frac{1}{3}a^{9}-\frac{23}{9}a^{8}+\frac{1}{3}a^{7}+\frac{19}{9}a^{6}+\frac{46}{9}a^{5}-\frac{1}{3}a^{4}-\frac{10}{3}a^{3}-\frac{20}{9}a^{2}-\frac{2}{9}a+\frac{11}{9}$, $\frac{11}{9}a^{17}-\frac{5}{9}a^{16}-a^{14}+\frac{2}{9}a^{13}-\frac{1}{9}a^{12}+\frac{19}{9}a^{11}-\frac{7}{9}a^{10}-\frac{2}{9}a^{9}-\frac{32}{3}a^{8}+\frac{41}{9}a^{7}-\frac{1}{3}a^{6}+\frac{113}{9}a^{5}-\frac{31}{9}a^{4}+\frac{1}{3}a^{3}-\frac{25}{9}a^{2}-\frac{4}{9}a+\frac{2}{9}$, $\frac{2}{9}a^{16}+\frac{5}{9}a^{15}-\frac{1}{3}a^{14}+\frac{1}{9}a^{13}-\frac{4}{9}a^{12}-\frac{1}{9}a^{11}+\frac{1}{3}a^{10}+\frac{8}{9}a^{9}-\frac{7}{9}a^{8}-\frac{4}{3}a^{7}-5a^{6}+\frac{19}{9}a^{5}+\frac{2}{9}a^{4}+\frac{47}{9}a^{3}-\frac{7}{9}a^{2}+a-\frac{14}{9}$ | 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: | \( 153.95173876 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 153.95173876 \cdot 1}{6\cdot\sqrt{5040742784941043712}}\cr\approx \mathstrut & 0.17442345891 \end{aligned}\]
Galois group
$C_3^2:C_6$ (as 18T23):
A solvable group of order 54 |
The 18 conjugacy class representatives for $C_3^2:C_6$ |
Character table for $C_3^2:C_6$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.1714608.2, 6.0.107163.1, 6.0.21168.1, 6.0.34992.1 |
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 27 sibling: | 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 | R | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{9}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
\(3\) | Deg $18$ | $6$ | $3$ | $21$ | |||
\(7\) | 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |