Normalized defining polynomial
\( x^{18} - 3x^{15} + 15x^{12} + 20x^{9} + 33x^{6} + 6x^{3} + 1 \)
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: | \(-4052555153018976267\) \(\medspace = -\,3^{39}\) | 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: | \(10.81\) | 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: | $3^{13/6}\approx 10.808432596584025$ | ||
Ramified primes: | \(3\) | 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: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{2}-\frac{1}{9}a+\frac{1}{9}$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{9}+\frac{1}{9}a^{3}+\frac{1}{9}$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{10}+\frac{1}{9}a^{4}+\frac{1}{9}a$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{9}a^{5}+\frac{1}{9}a-\frac{1}{9}$, $\frac{1}{171}a^{15}-\frac{5}{171}a^{12}+\frac{2}{57}a^{9}-\frac{11}{171}a^{6}-\frac{2}{171}a^{3}+\frac{16}{57}$, $\frac{1}{513}a^{16}+\frac{1}{513}a^{15}+\frac{14}{513}a^{13}+\frac{14}{513}a^{12}+\frac{25}{513}a^{10}+\frac{25}{513}a^{9}+\frac{46}{513}a^{7}+\frac{46}{513}a^{6}+\frac{131}{513}a^{4}+\frac{131}{513}a^{3}+\frac{124}{513}a+\frac{124}{513}$, $\frac{1}{513}a^{17}-\frac{1}{513}a^{15}+\frac{14}{513}a^{14}-\frac{14}{513}a^{12}+\frac{25}{513}a^{11}-\frac{25}{513}a^{9}+\frac{46}{513}a^{8}-\frac{46}{513}a^{6}+\frac{131}{513}a^{5}-\frac{131}{513}a^{3}+\frac{124}{513}a^{2}-\frac{124}{513}$
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: | \( \frac{31}{171} a^{15} - \frac{98}{171} a^{12} + \frac{157}{57} a^{9} + \frac{571}{171} a^{6} + \frac{793}{171} a^{3} - \frac{17}{57} \) (order $18$) | 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: | $\frac{31}{171}a^{16}-\frac{98}{171}a^{13}+\frac{157}{57}a^{10}+\frac{571}{171}a^{7}+\frac{793}{171}a^{4}-\frac{17}{57}a$, $\frac{47}{171}a^{16}-\frac{53}{57}a^{13}+\frac{757}{171}a^{10}+\frac{680}{171}a^{7}+\frac{131}{19}a^{4}-\frac{347}{171}a$, $\frac{67}{171}a^{17}+\frac{104}{513}a^{16}-\frac{49}{513}a^{15}-\frac{202}{171}a^{14}-\frac{368}{513}a^{13}+\frac{169}{513}a^{12}+\frac{1010}{171}a^{11}+\frac{1745}{513}a^{10}-\frac{826}{513}a^{9}+\frac{1315}{171}a^{8}+\frac{1193}{513}a^{7}-\frac{544}{513}a^{6}+\frac{2222}{171}a^{5}+\frac{2566}{513}a^{4}-\frac{1631}{513}a^{3}+\frac{404}{171}a^{2}-\frac{784}{513}a+\frac{137}{513}$, $\frac{5}{171}a^{17}-\frac{127}{513}a^{16}-\frac{40}{513}a^{15}-\frac{25}{171}a^{14}+\frac{388}{513}a^{13}+\frac{124}{513}a^{12}+\frac{106}{171}a^{11}-\frac{1921}{513}a^{10}-\frac{601}{513}a^{9}-\frac{55}{171}a^{8}-\frac{2422}{513}a^{7}-\frac{814}{513}a^{6}-\frac{10}{171}a^{5}-\frac{4040}{513}a^{4}-\frac{965}{513}a^{3}-\frac{368}{171}a^{2}-\frac{301}{513}a+\frac{56}{513}$, $\frac{55}{513}a^{17}+\frac{86}{513}a^{15}-\frac{199}{513}a^{14}-\frac{278}{513}a^{12}+\frac{919}{513}a^{11}+\frac{1352}{513}a^{9}+\frac{649}{513}a^{8}+\frac{1391}{513}a^{6}+\frac{935}{513}a^{5}+\frac{2602}{513}a^{3}-\frac{647}{513}a^{2}+\frac{119}{513}$, $\frac{37}{513}a^{17}-\frac{34}{513}a^{16}-\frac{71}{513}a^{15}-\frac{109}{513}a^{14}+\frac{94}{513}a^{13}+\frac{203}{513}a^{12}+\frac{526}{513}a^{11}-\frac{508}{513}a^{10}-\frac{1034}{513}a^{9}+\frac{847}{513}a^{8}-\frac{709}{513}a^{7}-\frac{1556}{513}a^{6}+\frac{971}{513}a^{5}-\frac{1661}{513}a^{4}-\frac{2632}{513}a^{3}-\frac{257}{513}a^{2}-\frac{454}{513}a-\frac{197}{513}$, $\frac{67}{171}a^{17}+\frac{40}{513}a^{16}+\frac{37}{513}a^{15}-\frac{202}{171}a^{14}-\frac{124}{513}a^{13}-\frac{109}{513}a^{12}+\frac{1010}{171}a^{11}+\frac{601}{513}a^{10}+\frac{526}{513}a^{9}+\frac{1315}{171}a^{8}+\frac{814}{513}a^{7}+\frac{847}{513}a^{6}+\frac{2222}{171}a^{5}+\frac{965}{513}a^{4}+\frac{971}{513}a^{3}+\frac{404}{171}a^{2}+\frac{457}{513}a-\frac{257}{513}$, $\frac{8}{19}a^{17}+\frac{89}{513}a^{16}-\frac{1}{27}a^{15}-\frac{227}{171}a^{14}-\frac{293}{513}a^{13}+\frac{4}{27}a^{12}+\frac{124}{19}a^{11}+\frac{1427}{513}a^{10}-\frac{19}{27}a^{9}+\frac{140}{19}a^{8}+\frac{1358}{513}a^{7}-\frac{1}{27}a^{6}+\frac{2212}{171}a^{5}+\frac{2596}{513}a^{4}-\frac{23}{27}a^{3}+\frac{4}{19}a^{2}-\frac{193}{513}a+\frac{8}{27}$ | 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: | \( 529.942929585 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 529.942929585 \cdot 1}{18\cdot\sqrt{4052555153018976267}}\cr\approx \mathstrut & 0.223208611212 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 18T3):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3 \times C_3$ |
Character table for $S_3 \times C_3$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, \(\Q(\zeta_{9})^+\), 6.0.177147.2, 6.0.177147.1 x2, \(\Q(\zeta_{9})\), 9.3.1162261467.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 sibling: | 6.0.177147.1 |
Degree 9 sibling: | 9.3.1162261467.1 |
Minimal sibling: | 6.0.177147.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $18$ | $18$ | $1$ | $39$ |