Normalized defining polynomial
\( x^{18} - 36 x^{16} + 540 x^{14} - 4368 x^{12} + 20592 x^{10} - 57024 x^{8} + 88704 x^{6} - 69120 x^{4} + \cdots - 1536 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(396521139274783615537700143104\) \(\medspace = 2^{27}\cdot 3^{45}\) | 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: | \(44.09\) | 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: | $2^{3/2}3^{5/2}\approx 44.090815370097204$ | ||
Ramified primes: | \(2\), \(3\) | 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: | \(\Q(\sqrt{6}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(216=2^{3}\cdot 3^{3}\) | ||
Dirichlet character group: | $\lbrace$$\chi_{216}(1,·)$, $\chi_{216}(131,·)$, $\chi_{216}(193,·)$, $\chi_{216}(73,·)$, $\chi_{216}(11,·)$, $\chi_{216}(145,·)$, $\chi_{216}(83,·)$, $\chi_{216}(203,·)$, $\chi_{216}(25,·)$, $\chi_{216}(155,·)$, $\chi_{216}(97,·)$, $\chi_{216}(35,·)$, $\chi_{216}(169,·)$, $\chi_{216}(107,·)$, $\chi_{216}(49,·)$, $\chi_{216}(179,·)$, $\chi_{216}(121,·)$, $\chi_{216}(59,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | 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: | $\frac{1}{64}a^{12}-\frac{3}{8}a^{10}+\frac{27}{8}a^{8}-14a^{6}+\frac{105}{4}a^{4}-18a^{2}+2$, $\frac{1}{8}a^{6}-\frac{3}{2}a^{4}+\frac{9}{2}a^{2}-1$, $\frac{1}{2}a^{2}-2$, $\frac{1}{128}a^{14}-\frac{15}{64}a^{12}+\frac{45}{16}a^{10}-\frac{275}{16}a^{8}+\frac{449}{8}a^{6}-\frac{371}{4}a^{4}+\frac{125}{2}a^{2}-5$, $\frac{1}{4}a^{4}-2a^{2}+2$, $\frac{1}{128}a^{14}-\frac{7}{32}a^{12}+\frac{77}{32}a^{10}-\frac{105}{8}a^{8}+\frac{147}{4}a^{6}-49a^{4}+\frac{49}{2}a^{2}-2$, $\frac{1}{16}a^{8}-\frac{7}{8}a^{6}+\frac{15}{4}a^{4}-5a^{2}+1$, $\frac{1}{16}a^{8}-a^{6}+5a^{4}-8a^{2}+2$, $\frac{1}{128}a^{15}-\frac{15}{64}a^{13}-\frac{1}{64}a^{12}+\frac{45}{16}a^{11}+\frac{3}{8}a^{10}-\frac{275}{16}a^{9}-\frac{27}{8}a^{8}+\frac{225}{4}a^{7}+14a^{6}-\frac{189}{2}a^{5}-\frac{105}{4}a^{4}+\frac{139}{2}a^{3}+18a^{2}-12a-1$, $\frac{1}{128}a^{14}-\frac{7}{32}a^{12}+\frac{77}{32}a^{10}-\frac{105}{8}a^{8}-\frac{1}{8}a^{7}+\frac{147}{4}a^{6}+\frac{7}{4}a^{5}-49a^{4}-7a^{3}+\frac{49}{2}a^{2}+7a-1$, $\frac{1}{8}a^{6}-\frac{3}{2}a^{4}-\frac{1}{2}a^{3}+\frac{9}{2}a^{2}+3a-1$, $\frac{1}{128}a^{14}-\frac{7}{32}a^{12}+\frac{1}{32}a^{11}+\frac{77}{32}a^{10}-\frac{11}{16}a^{9}-\frac{105}{8}a^{8}+\frac{11}{2}a^{7}+\frac{147}{4}a^{6}-\frac{77}{4}a^{5}-\frac{197}{4}a^{4}+\frac{55}{2}a^{3}+\frac{53}{2}a^{2}-11a-5$, $\frac{1}{32}a^{11}-\frac{11}{16}a^{9}+\frac{43}{8}a^{7}-\frac{35}{2}a^{5}+\frac{1}{4}a^{4}+\frac{41}{2}a^{3}-2a^{2}-4a+1$, $\frac{1}{32}a^{10}-\frac{5}{8}a^{8}+\frac{35}{8}a^{6}-\frac{1}{4}a^{5}-\frac{25}{2}a^{4}+\frac{5}{2}a^{3}+\frac{25}{2}a^{2}-5a-1$, $\frac{1}{64}a^{13}-\frac{13}{32}a^{11}+\frac{65}{16}a^{9}+\frac{1}{16}a^{8}-\frac{39}{2}a^{7}-a^{6}+\frac{181}{4}a^{5}+5a^{4}-43a^{3}-8a^{2}+8a+1$, $\frac{1}{2}a^{2}-a-1$, $\frac{1}{256}a^{17}-\frac{1}{256}a^{16}-\frac{17}{128}a^{15}+\frac{1}{8}a^{14}+\frac{119}{64}a^{13}-\frac{13}{8}a^{12}-\frac{221}{16}a^{11}+11a^{10}+\frac{935}{16}a^{9}-\frac{165}{4}a^{8}-\frac{561}{4}a^{7}+84a^{6}+\frac{357}{2}a^{5}-84a^{4}-102a^{3}+32a^{2}+16a-1$ (assuming GRH) | 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: | \( 1186002844.94 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 1186002844.94 \cdot 1}{2\cdot\sqrt{396521139274783615537700143104}}\cr\approx \mathstrut & 0.246866684536 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 18 |
The 18 conjugacy class representatives for $C_{18}$ |
Character table for $C_{18}$ |
Intermediate fields
\(\Q(\sqrt{6}) \), \(\Q(\zeta_{9})^+\), 6.6.10077696.1, \(\Q(\zeta_{27})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | $18$ | $18$ | $18$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{18}$ | $18$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.18.27.118 | $x^{18} - 20 x^{17} + 262 x^{16} - 448 x^{15} - 6176 x^{14} + 33472 x^{13} + 381184 x^{12} + 1775360 x^{11} - 26524896 x^{10} - 40518016 x^{9} + 250659264 x^{8} + 956865536 x^{7} - 263783424 x^{6} - 3209602048 x^{5} - 5614871552 x^{4} - 16485806080 x^{3} - 37256698624 x^{2} - 144388781056 x - 459719889408$ | $2$ | $9$ | $27$ | $C_{18}$ | $[3]^{9}$ |
\(3\) | Deg $18$ | $18$ | $1$ | $45$ |